Instability of spherical naked singularities of a scalar field under gravitational perturbations

Junbin Li Department of Mathematics, Sun Yat-sen University
Guangzhou, China
lijunbin@mail.sysu.edu.cn
 and  Jue Liu Department of Mathematics, Sun Yat-sen University
Guangzhou, China
liuj337@mail2.sysu.edu.cn
Abstract.

In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is shown that, arbitrarily fixing the initial scalar field, the set of the initial conformal metrics on the outgoing null cone such that the maximal future development does not have any sequences of closed trapped surfaces approaching the singularity, is of first category in the whole space in which the shear tensors are continuous. Such a set can then be viewed as exceptional, although the exceptionality is weaker than the at least 111 co-dimensionality in spherical symmetry. Almost equivalently, it is also proved that, arbitrarily fixing an incoming null cone C¯εsubscript¯𝐶𝜀\underline{C}_{\varepsilon} to the future of the initial incoming null cone, the set of the initial conformal metrics such that the maximal future development has at least one closed trapped surface before C¯εsubscript¯𝐶𝜀\underline{C}_{\varepsilon}, contains an open and dense subset of the whole space. Since the initial scalar field can be chosen such that the singularity is naked if the initial shear is set to be zero, we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations. This in particular gives new families of non-spherically symmetric gravitational perturbations different from the original spherically symmetric scalar perturbations given by Christodoulou.

1. Introduction

1.1. Previous works

In general relativity, one models the gravity in spacetimes using a Lorentzian manifold whose metric should satisfy the Einstein equations

𝐑𝐢𝐜αβ12𝐑gαβ=𝐓αβ.subscript𝐑𝐢𝐜𝛼𝛽12𝐑subscript𝑔𝛼𝛽subscript𝐓𝛼𝛽\mathbf{Ric}_{\alpha\beta}-\frac{1}{2}\mathbf{R}g_{\alpha\beta}=\mathbf{T}_{\alpha\beta}.

One fundamental question in the classical theory of general relativity is the weak cosmic censorship conjecture, which is proposed firstly by Penrose, and usually stated as follows: for suitable Einstein-matter field systems, the maximal future development of the generic asymptotically flat initial data possesses a complete future null infinity.

Perhaps the most interesting case is the vacuum Einstein equations, i.e., setting 𝐓αβ0subscript𝐓𝛼𝛽0\mathbf{T}_{\alpha\beta}\equiv 0 and hence

𝐑𝐢𝐜αβ=0,subscript𝐑𝐢𝐜𝛼𝛽0\displaystyle\mathbf{Ric}_{\alpha\beta}=0,

This is because the singularities arise in this case only because of the effect of gravity itself. In order to simplify the equations, symmetries are usually imposed, for example, spherical symmetry. However, it is not appropriate to impose spherical symmetry in vacuum because the spacetime will reduce to a single family of the static Schwarzschild solutions due to the Birkhoff theorem. To gain insights to attack the original problem, one would rather investigate the spherically symmetric solution of the Einstein equations coupled with a simple material model that can be used to simulate the effect of gravity. A suitable choice is the massless scalar field ϕitalic-ϕ\phi, whose energy-momentum tensor reads

𝐓αβ=αϕβϕ12gαβμϕμϕ.subscript𝐓𝛼𝛽subscript𝛼italic-ϕsubscript𝛽italic-ϕ12subscript𝑔𝛼𝛽subscript𝜇italic-ϕsuperscript𝜇italic-ϕ\mathbf{T}_{\alpha\beta}=\nabla_{\alpha}\phi\nabla_{\beta}\phi-\frac{1}{2}g_{\alpha\beta}\nabla_{\mu}\phi\nabla^{\mu}\phi.

The coupled system, which we call the Einstein-scalar field equations, can be written in the following form:

{𝐑𝐢𝐜αβ=2αϕβϕgαβαβϕ=0.casessubscript𝐑𝐢𝐜𝛼𝛽2subscript𝛼italic-ϕsubscript𝛽italic-ϕotherwisesuperscript𝑔𝛼𝛽subscript𝛼subscript𝛽italic-ϕ0otherwise\begin{cases}\mathbf{Ric}_{\alpha\beta}=2\nabla_{\alpha}\phi\nabla_{\beta}\phi\\ g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi=0\end{cases}.

The study of this model in spherical symmetry is successful. Christodoulou studied in a series of papers, including [3, 4, 5, 6, 8], the spherical symmetric solution of this model, and rigorously verified in the last paper [8] that the weak cosmic censorship conjecture holds in this category. Strictly speaking, he proved the following genericity theorem.

Theorem 1.1 (Christodoulou [3, 4, 5, 8]).

Let 𝒜𝒜\mathcal{A} be the space of functions of bounded variation on the non-negative real line, which is the space of the initial data α0=r(rϕ)|Cosubscript𝛼0evaluated-at𝑟𝑟italic-ϕsubscript𝐶𝑜\alpha_{0}=\frac{\partial}{\partial r}(r\phi)\Big{|}_{C_{o}} on the initial null cone Cosubscript𝐶𝑜C_{o} issuing from a point o𝑜o, a fixed point of the SO(3)𝑆𝑂3SO(3) action, where r𝑟r is the area radius of the orbit spherical sections of Cosubscript𝐶𝑜C_{o}. Let 𝒮𝒮\mathcal{S} be the collection of α0𝒜subscript𝛼0𝒜\alpha_{0}\in\mathcal{A} such that the maximal future development is not complete. Let \mathcal{E} be the exceptional set which is the complement in 𝒮𝒮\mathcal{S} of the collection of initial data such that the maximal development has a complete future null infinity and terminates at a spacelike singular future boundary. Then if α0subscript𝛼0\alpha_{0}\in\mathcal{E}, there exists some f𝒜𝑓𝒜f\in\mathcal{A} depending on α0subscript𝛼0\alpha_{0} such that the family α0+tfsubscript𝛼0𝑡𝑓\alpha_{0}+tf lies in 𝒮\\𝒮\mathcal{S}\backslash\mathcal{E} for t0𝑡0t\neq 0. In addition, any two such families do not intersect, i.e., if α1+t1f1=α2+t2f2subscript𝛼1subscript𝑡1subscript𝑓1subscript𝛼2subscript𝑡2subscript𝑓2\alpha_{1}+t_{1}f_{1}=\alpha_{2}+t_{2}f_{2}, then α1α2subscript𝛼1subscript𝛼2\alpha_{1}\equiv\alpha_{2}, f1f2subscript𝑓1subscript𝑓2f_{1}\equiv f_{2}, t1=t2subscript𝑡1subscript𝑡2t_{1}=t_{2}.

The conclusion of the above theorem consists of two parts: the existence of the families (α0+tf)tsubscriptsubscript𝛼0𝑡𝑓𝑡(\alpha_{0}+tf)_{t\in\mathbb{R}} of perburbations, which implies that \mathcal{E} is dense in 𝒜𝒜\mathcal{A}, and the uniqueness, which means that any functions in 𝒜𝒜\mathcal{A} belongs to no more than one of such families. The set \mathcal{E} is then of co-dimension at least 111111In fact, Christodoulou proved that indeed the co-dimension is at least 222. Christodoulou’s proof also suggested that the set \mathcal{E} still has at least 111 co-dimension in the space of all more regular absolutely continuous initial data. and therefore the set \mathcal{E} can be considered to be exceptional. Indeed, this theorem does not only establish the weak cosmic censorship but also the strong cosmic censorship, which says that the maximal future development of generic asymptotically flat initial data cannot extend as a Lorentzian manifold in suitable sense. We remark that the exceptional set \mathcal{E} is not empty, i.e., singularities not hidden inside a black hole, which we call naked singularities, do occur in the spherically symmetric solutions of the Einstein-scalar field equations. Examples are provided by Christodoulou in [6] and this shows that the word “generic” is needed in both the statements of the weak and strong cosmic censorship conjectures.

An essential step of proving the above theorem is to understand in which way a closed trapped surface, a 2-spacelike surface embedding in the spacetime such that the mean curvature relative to both future null normals are negetive forms surrounding a given singularity. This was accomplished by Christodoulou in [4]:

Theorem 1.2 (Christodoulou, [4]).

Consider the spherically symmetric solution of the Einstein-scalar field equations with initial data given on a null cone Cosubscript𝐶𝑜C_{o}. Consider two spherical sections S1subscript𝑆1S_{1} and S2subscript𝑆2S_{2} with area radii r1subscript𝑟1r_{1}, r2subscript𝑟2r_{2} and mass contents m1subscript𝑚1m_{1}, m2subscript𝑚2m_{2}, and S2subscript𝑆2S_{2} is in the exterior to S1subscript𝑆1S_{1}. Denote

δ=r2r11.𝛿subscript𝑟2subscript𝑟11\delta=\frac{r_{2}}{r_{1}}-1.

Then there exists positive constants c0subscript𝑐0c_{0}, c1subscript𝑐1c_{1} such that if δc0𝛿subscript𝑐0\delta\leq c_{0} and

2(m2m1)>c1r2δlog(1δ),2subscript𝑚2subscript𝑚1subscript𝑐1subscript𝑟2𝛿1𝛿\displaystyle 2(m_{2}-m_{1})>c_{1}r_{2}\delta\log\left(\frac{1}{\delta}\right), (1.1)

then there is a closed trapped surface on the incoming null cone through S2subscript𝑆2S_{2} in the maximal future development.

Remark 1.1.

Theorem 1.2 is in fact part of the main theorem called collapse theorem in [4], in which the behaviors of the singularity and the event horizon were also studied. First, it was proved that region of trapped surfaces terminates at a strictly spacelike singular boundary. Second, it was also proved that the event horizon will form and the completeness of the future null infinity follows from the analysis in [3], that every causal curve r=c𝑟𝑐r=c has infinite length towards the future for c𝑐c larger than the final Bondi mass of the spacetime.222It was also discussed in [11] by Dafermos how a single closed trapped surface implies the completeness of the future null infinity in spherically symmetric spacetimes. Beyond spherical symmetry, nothing is known about the relation between the existence of a closed trapped surface and the completeness of the future null infinity. We would like to mention however the works [17, 18] on the local existence theorems in retarded time, while the completeness of the future null infinity is equivalent to the global existence in retarded time.

The proof of Theorem 1.1 can then be sketched as follows. It was shown in [5] that the first singularity can only appear on the central line. Suppose that e𝑒e is this singular endpoint of the central line. We define C¯esubscript¯𝐶𝑒\underline{C}_{e} to be the boundary of the causal past of e𝑒e, whose intersection with Cosubscript𝐶𝑜C_{o} has area radius resubscript𝑟𝑒r_{e}. Then what was actually proved in the last paper [8] is the following instability theorem:

Theorem 1.3 (Christodoulou, [8]).

Suppose that the initial data α0subscript𝛼0\alpha_{0} satisfies some generic condition. Then there exist a sequence of rnre+subscript𝑟𝑛superscriptsubscript𝑟𝑒r_{n}\to r_{e}^{+} and a sequence of points pnesubscript𝑝𝑛𝑒p_{n}\to e on the central line, such that the outgoing null cone Cpnsubscript𝐶subscript𝑝𝑛C_{p_{n}} issuing from pnsubscript𝑝𝑛p_{n} satisfies the assumptions of Theorem 1.2 at two spherical sections: S1,nsubscript𝑆1𝑛S_{1,n}, the intersection of Cpnsubscript𝐶subscript𝑝𝑛C_{p_{n}} and C¯esubscript¯𝐶𝑒\underline{C}_{e}, and S2,nsubscript𝑆2𝑛S_{2,n}, the intersection of Cpnsubscript𝐶subscript𝑝𝑛C_{p_{n}} and C¯rnsubscript¯𝐶subscript𝑟𝑛\underline{C}_{r_{n}}, where C¯rnsubscript¯𝐶subscript𝑟𝑛\underline{C}_{r_{n}} is the incoming null cone through the spherical section r=rn𝑟subscript𝑟𝑛r=r_{n} on the initial null cone Cosubscript𝐶𝑜C_{o}.

Remark 1.2.

When we say that a condition is generic, it means that in a certain sense, almost all initial data satisfy this condition. It was also verified in [8] that the set of initial data not satisfying the generic condition in Theorem 1.3 is of co-dimension at least 111 in the space of all initial data.

As a consequence by applying Theorem 1.2, there exists a sequence of closed trapped surfaces. They are the orbit spheres, approaching the singularity e𝑒e and their areas tend to zero. Then the apparent horizon issues from e𝑒e and Theorem 1.1 follows immediately from the conclusions mentioned in Remark 1.1.

Inspired by the above argument, Christodoulou formulated a conjecture [7] for the vacuum Einstein equations, which can be termed the trapped surface conjecture, and be viewed as a local version of the weak cosmic censorship conjecture and part of the strong one. It can of course be viewed as a conjecture for the Einstein equations coupled with suitable matter field without any modifications.

Conjecture (Christodouou, [7]).

The maximal future development (,g)𝑔(\mathcal{M},g) of generic asymptotically flat initial data (,g¯,k)¯𝑔𝑘(\mathcal{H},\bar{g},k) has the following property. If P𝑃P is a TIP333TIP is short for the terminal indecomposable past set, which is originally introduced in [12]. Roughly speaking, a TIP is the past of a piece of the singular boundaries of the maximal future development. in M𝑀M whose trace on \mathcal{H} has compact closure 𝒦𝒦\mathcal{K}, then for any open domain 𝒟𝒟\mathcal{D} in \mathcal{H} containing 𝒦𝒦\mathcal{K}, the domain of dependence of 𝒟𝒟\mathcal{D} in \mathcal{M} contains a closed trapped surface.

The goal of this paper is to initiate the study of the above conjecture without symmetries, or precisely speaking, study how the arguments of proving Theorem 1.1 can be generalized when no symmetries are imposed.

1.2. Main results

In this paper, we study the next simpliest case that the TIP is the past spherical singularities of a massless scalar field studied in [8]. The meaning of a singularity being spherical in this paper is that the causal past of this singularity is a spherically symmetric spacetime and the boundary of this causal past is foliated by orbit spheres. In contrast to the spherical symmetry imposed on the causal past of the singularity, no symmetries are required in the future of the boundary of this causal past. We expect we can gain some insights to attack the problem when the singularity is not assumed to be spherical.

Recalling that in spherical symmetry, a sharp criterion of a point e𝑒e on the central line being singular is obtained in [5]. It says that if the ratio of the Hawking mass to the radius of the spheres does not tend to zero as we approach e𝑒e from the past, then the regular solution cannot extend across e𝑒e. Then the question we study is formulated as follows.

Basic setup.

In this paper, we study the following characteristic initial value problem with a spherical singularity of a scalar field. The initial data is given on two intersecting null cone Cu0subscript𝐶subscript𝑢0C_{u_{0}} which is outgoing and C¯0subscript¯𝐶0\underline{C}_{0} which is incoming. The data on C¯0subscript¯𝐶0\underline{C}_{0} is assumed to be spherically symmetric. Denote r𝑟r to be the area radius of the spherical section on C¯0subscript¯𝐶0\underline{C}_{0} and m=m(r)𝑚𝑚𝑟m=m(r) be the Hawking mass. For r>0𝑟0r>0, it is assumed that r>2m0𝑟2𝑚0r>2m\geq 0 so no closed trapped surfaces exist on C¯0subscript¯𝐶0\underline{C}_{0} and 2mr02𝑚𝑟0\frac{2m}{r}\nrightarrow 0 as r0+𝑟superscript0r\to 0^{+}. The outgoing null cone Cu0subscript𝐶subscript𝑢0C_{u_{0}} intersects C¯0subscript¯𝐶0\underline{C}_{0} at r=u0𝑟subscript𝑢0r=-u_{0}. No symmetries are imposed on the data on Cu0subscript𝐶subscript𝑢0C_{u_{0}}. We then solve the Einstein-scalar field equations of such data. This setup is depicted in Figure 1.

Refer to caption
Figure 1. Basic setup

In this setup, we are able to prove some instability theorems, generalizing Theorem 1.3, which will be discussed in Section 1.5. Based on the instability theorems, we are able to prove the genericity theorem. Recall that the initial data set on Cu0subscript𝐶subscript𝑢0C_{u_{0}} consists of the conformal metric g/^\widehat{\mbox{$g\mkern-9.0mu/$}} of Cu0subscript𝐶subscript𝑢0C_{u_{0}}, the lapse ΩΩ\Omega and the scalar field function ϕitalic-ϕ\phi. For simplicity, we will fix smooth the initial data on C¯0subscript¯𝐶0\underline{C}_{0} and the smooth lapse ΩΩ\Omega and scalar field ϕitalic-ϕ\phi on Cu0subscript𝐶subscript𝑢0C_{u_{0}}. One can certainly state and prove a complete version allowing perturbations both on the g/^\widehat{\mbox{$g\mkern-9.0mu/$}} and ϕitalic-ϕ\phi without essential additional difficulties. We also let (u¯,u,ϑ)¯𝑢𝑢italic-ϑ(\underline{u},u,\vartheta) be the double null coordinate system introduced in Section 2.1, and then the initial quantities are functions of three variables (u¯,ϑ)¯𝑢italic-ϑ(\underline{u},\vartheta). The initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} is not assumed to be smooth. We present here a rough form of the main result of the present article, whose precise version is given in Theorem 6.3.

Theorem 1.4 (Main result).

Let the initial data on C¯0subscript¯𝐶0\underline{C}_{0} with a singular vertex and initial ΩΩ\Omega and Lϕ𝐿italic-ϕL\phi on Cu0subscript𝐶subscript𝑢0C_{u_{0}} be arbitrarily smoothly fixed. Let \mathcal{I} be all initial conformal metrics g/^\widehat{\mbox{$g\mkern-9.0mu/$}} defined on Cu0subscript𝐶subscript𝑢0C_{u_{0}} for 0u¯10¯𝑢10\leq\underline{u}\leq 1 such that g/^Cu¯1Hϑ10\widehat{\mbox{$g\mkern-9.0mu/$}}\in C^{1}_{\underline{u}}H^{10}_{\vartheta} up to the intersection Cu0C¯0subscript𝐶subscript𝑢0subscript¯𝐶0C_{u_{0}}\cap\underline{C}_{0}444This means that g/^\widehat{\mbox{$g\mkern-9.0mu/$}} at u¯=0¯𝑢0\underline{u}=0 must be coincide in Cu¯1superscriptsubscript𝐶¯𝑢1C_{\underline{u}}^{1} level with the data induced by the spherical symmetric initial data on C¯0subscript¯𝐶0\underline{C}_{0}. This in particular implies that g/^\widehat{\mbox{$g\mkern-9.0mu/$}} is standard round metric and the shear χ^=0^𝜒0\widehat{\chi}=0 at u¯=0¯𝑢0\underline{u}=0.. Let \mathcal{E} be the set of g/^\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I} such that the future maximal development does not have any sequences of closed trapped surfaces approaching the singularity. Then \mathcal{E} is of first category in \mathcal{I}, i.e., csuperscript𝑐\mathcal{E}^{c}, the complement of \mathcal{E} in \mathcal{I}, contains a subset that is a countably intersection of open and dense subsets of \mathcal{I}.

Remark 1.3.

Although in a weaker sense than the at least 111 co-dimensionality proved in spherical symmetry, we can still say the set \mathcal{E} is exceptional, or the initial conformal metrics lying in \\\mathcal{I}\backslash\mathcal{E} are generic. What we have proved is essentially the existence of the perturbations. To prove the uniqueness part, one will need some fully anisotropic instability theorems which is not proved in this paper. This is one of the main differences between spherical symmetry and non-spherical symmetry. We will try to study this in the future works.

Remark 1.4.

The topology of the space of the initial conformal metrics we work in is the most regular one in which the estimates are done based on the methods in this paper. For non-smooth initial data, it is not a priori clear what “future maximal development” means. We will discuss this in Section 1.5.

Remark 1.5.

In the above theorem, the initial scalar field can be chosen such that the singularity is naked if no gravitational fields are present, i.e., the shear tensor is set to be zero. Such examples do exist by [6]. We already know from [8] that the naked singularities are not stable under spherical scalar perturbations in the sense of being of at least 111 co-dimension. In this paper, we find new perturbations contributed from the gravitational fields and establish the instability in the sense of being of first category. Therefore we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations.

In Theorem 1.4, we investigate the exceptionality of the set \mathcal{E} where the future maximal developments of the initial conformal metrics in csuperscript𝑐\mathcal{E}^{c} has a sequence of closed trapped surfaces approaching the singularity. But if one only concerns about the weak cosmic censorship conjecture, then it seems that only one single closed trapped surface is sufficient. We then let C¯εsubscript¯𝐶𝜀\underline{C}_{\varepsilon} be the incoming null cone where u¯=ε¯𝑢𝜀\underline{u}=\varepsilon and εsubscript𝜀\mathcal{E}_{\varepsilon} be the set of the initial conformal metrics in \mathcal{I} such that the maximal future development before C¯εsubscript¯𝐶𝜀\underline{C}_{\varepsilon} has no closed trapped surfaces. Note that εsubscript𝜀\mathcal{E}_{\varepsilon}\subset\mathcal{E} for any ε>0𝜀0\varepsilon>0, and assuming the closed trapped surfaces are stable under small perturbations of the initial conformal metrics in the topology of \mathcal{I}, we will have the following theorem.

Theorem 1.5.

εcsuperscriptsubscript𝜀𝑐\mathcal{E}_{\varepsilon}^{c} contains a subset that is open and dense in \mathcal{I} for all ε>0𝜀0\varepsilon>0.

This in particular implies that generically in the sense of being open and dense, there is at least one closed trapped surface surrounding the singularity in the maximal future development. Moreover, assuming Theorem 1.5, Theorem 1.4 is true by noting that =iεisubscript𝑖subscriptsubscript𝜀𝑖\mathcal{E}=\bigcup_{i}\mathcal{E}_{\varepsilon_{i}} for any sequence εi0subscript𝜀𝑖0\varepsilon_{i}\to 0. We should remark that the above argument on the equivalence of Theorem 1.4 and 1.5 is not rigorous since we are not working in smooth solutions and the stability of closed trapped surfaces, which is of course true in more regular solutions by Cauchy stability, will not be proved in this paper. But this argument still illustrates the close relation between these two statements, which will be proved simultaneously in their precise forms in Theorem 6.3 and 6.4.

1.3. The incoming cone C0subscript𝐶0\uline{C}_{0} and the singularity

In this subsection, we derive some basic properties of various quantities on C¯0subscript¯𝐶0\underline{C}_{0} which will be used throughout the whole paper. Readers who are not familiar with the double null coordinate system should refer to the next section first, where all notations are introduced in detail.

Let (u¯,u,ϑ)¯𝑢𝑢italic-ϑ(\underline{u},u,\vartheta) be the double null coordinate system, and the optical function u𝑢u is defined such that u=r𝑢𝑟u=-r on C¯0subscript¯𝐶0\underline{C}_{0}. Then the vector field L¯¯𝐿\underline{L}, tangent to C¯u¯subscript¯𝐶¯𝑢\underline{C}_{\underline{u}}, preserving the double null foliation, will satisfy L¯=u¯𝐿𝑢\underline{L}=\frac{\partial}{\partial u} and L¯u=1¯𝐿𝑢1\underline{L}u=1 on C¯0subscript¯𝐶0\underline{C}_{0}. Therefore, the null expansion Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} along L¯¯𝐿\underline{L} on C¯0subscript¯𝐶0\underline{C}_{0}, simply equals to 2|u|2𝑢-\frac{2}{|u|}. Let Lsuperscript𝐿L^{\prime} be the null vector field tangent to Cusubscript𝐶𝑢C_{u}, with g(L¯,L)=2𝑔¯𝐿superscript𝐿2g(\underline{L},L^{\prime})=-2, and trχtrsuperscript𝜒\mathrm{tr}\chi^{\prime} be the null expansion relative to Lsuperscript𝐿L^{\prime}. Denote

h=h(u)=|u|2trχ|C¯0,𝑢evaluated-at𝑢2trsuperscript𝜒subscript¯𝐶0\displaystyle h=h(u)=\frac{|u|}{2}\cdot\mathrm{tr}\chi^{\prime}\Big{|}_{\underline{C}_{0}},

then by the definition of Hawking mass m=m(u)𝑚𝑚𝑢m=m(u), we have 12mr=h12𝑚𝑟1-\frac{2m}{r}=h. The condition r>2m0𝑟2𝑚0r>2m\geq 0 implies that 0<h1010<h\leq 1 for |u|>0𝑢0|u|>0. Denote also

Ω0=Ω0(u)=Ω|C¯0subscriptΩ0subscriptΩ0𝑢evaluated-atΩsubscript¯𝐶0\displaystyle\Omega_{0}=\Omega_{0}(u)=\Omega\Big{|}_{\underline{C}_{0}}

where ΩΩ\Omega is the lapse function. Ω0subscriptΩ0\Omega_{0} can also defined intrinsicly on C¯0subscript¯𝐶0\underline{C}_{0} by L¯L¯=2(L¯logΩ0)L¯subscript¯𝐿¯𝐿2¯𝐿subscriptΩ0¯𝐿\nabla_{\underline{L}}\underline{L}=2(\underline{L}\log\Omega_{0})\underline{L}. Then from the null structure equation for D¯(Ωtrχ)¯𝐷Ωtr𝜒\underline{D}(\Omega\mathrm{tr}\chi), we can derive

u(log(Ω02h))=1|u|(11h).𝑢superscriptsubscriptΩ021𝑢11\displaystyle\frac{\partial}{\partial u}\left(\log(\Omega_{0}^{2}h)\right)=\frac{1}{|u|}\left(1-\frac{1}{h}\right). (1.2)

This implies that Ω02hsuperscriptsubscriptΩ02\Omega_{0}^{2}h is monotonically decreasing, even when the vertex of C¯0subscript¯𝐶0\underline{C}_{0} is assumed to be regular. According to Lemma 2 in [8], that 2mr02𝑚𝑟0\frac{2m}{r}\nrightarrow 0 implies that

limu0u0u1|u|(1h1)du=+.subscript𝑢superscript0superscriptsubscriptsubscript𝑢0𝑢1superscript𝑢11differential-dsuperscript𝑢\displaystyle\lim_{u\to 0^{-}}\int_{u_{0}}^{u}\frac{1}{|u^{\prime}|}\left(\frac{1}{h}-1\right)\mathrm{d}u^{\prime}=+\infty.

Consequently, Ω02h0superscriptsubscriptΩ020\Omega_{0}^{2}h\to 0 as u0𝑢superscript0u\to 0^{-}. It is mentioned in [7] that the quantity log(Ω02h)superscriptsubscriptΩ02-\log(\Omega_{0}^{2}h) measures the blue shift of light received by e𝑒e. Similar to [8], this fact is crucial in proving the instability of naked singularities.

Now we consider the scalar field on C¯0subscript¯𝐶0\underline{C}_{0}. Denote

ψ=ψ(u)=|u|L¯ϕ|C¯0,φ=φ(u)=|u|Lϕ|C¯0formulae-sequence𝜓𝜓𝑢evaluated-at𝑢¯𝐿italic-ϕsubscript¯𝐶0𝜑𝜑𝑢evaluated-at𝑢𝐿italic-ϕsubscript¯𝐶0\displaystyle\psi=\psi(u)=|u|\underline{L}\phi\Big{|}_{\underline{C}_{0}},\ \varphi=\varphi(u)=|u|L\phi\Big{|}_{\underline{C}_{0}}

where L𝐿L tangent to Cusubscript𝐶𝑢C_{u} preserves the double null foliation, and satisfies g(L¯,L)=2Ω2𝑔¯𝐿𝐿2superscriptΩ2g(\underline{L},L)=-2\Omega^{2}, or L=Ω2L𝐿superscriptΩ2superscript𝐿L=\Omega^{2}L^{\prime}. Consider the Raychaudhuri equation on C¯0subscript¯𝐶0\underline{C}_{0}:

D¯(Ωtrχ¯)=12(Ωtrχ¯)2|Ωχ¯^|22(L¯ϕ)2+2ω¯Ωtrχ¯.¯𝐷Ωtr¯𝜒12superscriptΩtr¯𝜒2superscriptΩ^¯𝜒22superscript¯𝐿italic-ϕ22¯𝜔Ωtr¯𝜒\underline{D}(\Omega\mathrm{tr}\underline{\chi})=-\frac{1}{2}(\Omega\mathrm{tr}\underline{\chi})^{2}-|\Omega\widehat{\underline{\chi}}|^{2}-2(\underline{L}\phi)^{2}+2\underline{\omega}\Omega\mathrm{tr}\underline{\chi}.

Plugging in Ωtrχ¯=2|u|Ωtr¯𝜒2𝑢\Omega\mathrm{tr}\underline{\chi}=-\frac{2}{|u|}, we have

ulogΩ=ω¯=12ψ2|u|.𝑢Ω¯𝜔12superscript𝜓2𝑢\frac{\partial}{\partial u}\log\Omega=\underline{\omega}=-\frac{1}{2}\frac{\psi^{2}}{|u|}.

Integrating the above equation, we have

logΩ02(u)Ω02(u0)=u0uψ2|u|du.superscriptsubscriptΩ02𝑢superscriptsubscriptΩ02subscript𝑢0superscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢\displaystyle-\log\frac{\Omega_{0}^{2}(u)}{\Omega_{0}^{2}(u_{0})}=\int_{u_{0}}^{u}\frac{\psi^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}. (1.3)

This implies that Ω0subscriptΩ0\Omega_{0} is monotonically decreasing no matter whether the vertex of C¯0subscript¯𝐶0\underline{C}_{0} is singular. This is an important formula which is used to estimate L¯ϕ¯𝐿italic-ϕ\underline{L}\phi in the a priori estimates. We also claim that Ω02h0superscriptsubscriptΩ020\Omega_{0}^{2}h\to 0 implies Ω00subscriptΩ00\Omega_{0}\to 0 as u0𝑢superscript0u\to 0^{-}. If not, then Ω0subscriptΩ0\Omega_{0} has a lower bound because it is decreasing, and then Ω02hsuperscriptsubscriptΩ02\Omega_{0}^{2}h tending to zero implies that hh should tend to zero. However, from (1.2), we have |u|u(Ω02h)=Ω02(h1)𝑢subscript𝑢superscriptsubscriptΩ02superscriptsubscriptΩ021|u|\partial_{u}(\Omega_{0}^{2}h)=\Omega_{0}^{2}(h-1), which would imply Ω02hsuperscriptsubscriptΩ02\Omega_{0}^{2}h\to-\infty, a contradiction. The fact that Ω0subscriptΩ0\Omega_{0} tends to zero monotonically is used throughout this paper. In summarize,

Lemma 1.1.

On C¯0subscript¯𝐶0\underline{C}_{0}, Ω02hsuperscriptsubscriptΩ02\Omega_{0}^{2}h is monotonically decreasing approaching the vertex, and if in addition the vertex of C¯0subscript¯𝐶0\underline{C}_{0} is singular, then Ω02h0superscriptsubscriptΩ020\Omega_{0}^{2}h\to 0. The same conclusions hold for Ω0subscriptΩ0\Omega_{0}.

Remark 1.6.

The choice of u=r𝑢𝑟u=-r on C¯0subscript¯𝐶0\underline{C}_{0} determines Ω0subscriptΩ0\Omega_{0} up to a constant multiple. Therefore, throughout the paper, we will assume that Ω0(u0)1subscriptΩ0subscript𝑢01\Omega_{0}(u_{0})\leq 1 for simplicity. It follows from the monotonicity of Ω0subscriptΩ0\Omega_{0} that Ω0(u)1subscriptΩ0𝑢1\Omega_{0}(u)\leq 1 for all u[u0,0)𝑢subscript𝑢00u\in[u_{0},0).

At last, we investigate the function φ𝜑\varphi. In terms of ψ𝜓\psi and φ𝜑\varphi, the wave equation restricted on C¯0subscript¯𝐶0\underline{C}_{0} can be written in the following form:

uφ=Ω02h|u|ψ,𝑢𝜑superscriptsubscriptΩ02𝑢𝜓\frac{\partial}{\partial u}\varphi=-\frac{\Omega_{0}^{2}h}{|u|}\psi,

then we have

φ(u)=φ(u0)u0uΩ02hψ|u|du𝜑𝑢𝜑subscript𝑢0superscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02𝜓superscript𝑢differential-dsuperscript𝑢\displaystyle\varphi(u)=\varphi(u_{0})-\int_{u_{0}}^{u}\frac{\Omega_{0}^{2}h\psi}{|u^{\prime}|}\mathrm{d}u^{\prime}

The integral on the right hand side can be estimated by

u0uΩ02hψ|u|du(u0uΩ02|u|du)12(u0uΩ02|ψ|2|u|du)12Ω02(u0)|log|u||u0||12superscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02𝜓superscript𝑢differential-dsuperscript𝑢superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02superscript𝑢differential-dsuperscript𝑢12superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02superscript𝜓2superscript𝑢differential-dsuperscript𝑢12superscriptsubscriptΩ02subscript𝑢0superscript𝑢subscript𝑢012\begin{split}\int_{u_{0}}^{u}\frac{\Omega_{0}^{2}h\psi}{|u^{\prime}|}\mathrm{d}u^{\prime}&\leq\left(\int_{u_{0}}^{u}\frac{\Omega_{0}^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\left(\int_{u_{0}}^{u}\frac{\Omega_{0}^{2}|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\\ &\leq\Omega_{0}^{2}(u_{0})\left|\log{\frac{|u|}{|u_{0}|}}\right|^{\frac{1}{2}}\\ \end{split} (1.4)

where we use (1.3) to estimate u0uΩ02|ψ|2|u|duΩ02(u0)superscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02superscript𝜓2superscript𝑢differential-dsuperscript𝑢superscriptsubscriptΩ02subscript𝑢0\int_{u_{0}}^{u}\frac{\Omega_{0}^{2}|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\leq\Omega_{0}^{2}(u_{0}). This estimate is related to the fact that the lower bound assumption of m2m1subscript𝑚2subscript𝑚1m_{2}-m_{1} in Theorem 1.2 is sharp in general situations.

1.4. Formation of trapped surfaces

It is not difficult to imagine that in order to prove the genericity theorems, a theorem similar to Theorem 1.2 without symmetries should be proved. This is Theorem 5.2 in this paper. It is a corollary of Theorem 5.1, a more general form of the theorem on the formation of trapped surface. To illustrate the main ideas and difficulties we only present Theorem 5.2 here, which can also be stated as follows.

Theorem 1.6.

We assume the smooth data on C¯0subscript¯𝐶0\underline{C}_{0} is spherically symmetric but not necessarily singular at its vertex. Suppose also that the smooth initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} between two sections S1=S0,u0subscript𝑆1subscript𝑆0subscript𝑢0S_{1}=S_{0,u_{0}} and S2=Sδ,u0subscript𝑆2subscript𝑆𝛿subscript𝑢0S_{2}=S_{\delta,u_{0}} where δ𝛿\delta is a small parameter, satisfies an initial estimate555The scale invariant nsuperscript𝑛\mathbb{H}^{n} norm is defined in Section 3.:

max{supu0uu1|φ(u)|,|u0|sup0u¯δ(Ωχ^7(u¯,u0)+ω,Lϕ5(u¯,u0))}Ω02(u0)a|log|u1||u0||\max\left\{\sup_{u_{0}\leq u\leq u_{1}}|\varphi(u)|,|u_{0}|\sup_{0\leq\underline{u}\leq\delta}\left(\|\Omega\widehat{\chi}\|_{\mathbb{H}^{7}(\underline{u},u_{0})}+\|\omega,L\phi\|_{\mathbb{H}^{5}(\underline{u},u_{0})}\right)\right\}\leq\Omega_{0}^{2}(u_{0})a\left|\log\frac{|u_{1}|}{|u_{0}|}\right|

for some a1𝑎1a\geq 1, where u1(u0,0)subscript𝑢1subscript𝑢00u_{1}\in(u_{0},0) is defined by

Ω02(u1)|u1|=C2Ω04(u0)δa|log|u1||u0||subscriptsuperscriptΩ20subscript𝑢1subscript𝑢1superscript𝐶2superscriptsubscriptΩ04subscript𝑢0𝛿𝑎subscript𝑢1subscript𝑢0\Omega^{2}_{0}(u_{1})|u_{1}|=C^{2}\Omega_{0}^{4}(u_{0})\delta a\left|\log\frac{|u_{1}|}{|u_{0}|}\right| (1.5)

and such that Ω02(u0)|log|u1||u0||1superscriptsubscriptΩ02subscript𝑢0subscript𝑢1subscript𝑢01\Omega_{0}^{2}(u_{0})\left|\log\frac{|u_{1}|}{|u_{0}|}\right|\geq 1. Then there exists some universal large constant C1subscript𝐶1C_{1} such that the smooth solution of the Einstein-scalar field equations exists for 0u¯δ,u0uu1formulae-sequence0¯𝑢𝛿subscript𝑢0𝑢subscript𝑢10\leq\underline{u}\leq\delta,u_{0}\leq u\leq u_{1}. If in addition

infϑS20δ|u0|2(|Ωχ^|2+2|Lϕ|2)(u¯,u0,ϑ)du¯17C2Ω04(u0)δa|log|u1||u0||,subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0𝛿superscriptsubscript𝑢02superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢0italic-ϑdifferential-dsuperscript¯𝑢17superscript𝐶2superscriptsubscriptΩ04subscript𝑢0𝛿𝑎subscript𝑢1subscript𝑢0\inf_{\vartheta\in S^{2}}\int_{0}^{\delta}|u_{0}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u}^{\prime},u_{0},\vartheta)\mathrm{d}\underline{u}^{\prime}\geq 17C^{2}\Omega_{0}^{4}(u_{0})\delta a\left|\log\frac{|u_{1}|}{|u_{0}|}\right|, (1.6)

then Sδ,u1subscript𝑆𝛿subscript𝑢1S_{\delta,u_{1}} is a closed trapped surface.

The main difference of this theorem from Theorem 1.2 is that the initial data should satisfy some a priori bounds and this bound should appear in the lower bound assumption of the initial energy, and a claim on the existence of the solution is needed. This is because in spherically symmetric case, Theorem 1.2 is proved by contradiction, using monotonicity properties of the Einstein equations in spherical symmetry. These properties break down even when we consider a small perturbation of spherical symmetry. The main difficulty is now the existence of the solution deep into the place where a closed trapped surface has chance to form eventually. In order to overcome this difficulty, we should find a correct form of the a priori estimates which can only be derived by L2superscript𝐿2L^{2} based energy.

Remark 1.7.

As mentioned above, without symmetries, all theorems should be proved under some a priori bounds assumed on the initial data. However, no a priori bounds are assumed on the initial data on C¯0subscript¯𝐶0\underline{C}_{0}. We will explain this in Section 1.6.

It is worth mentioning that Theorem 1.6 should be very carefully written down because only the sharp form of such a theorem can be used to prove the Theorem 1.4. In particular, the dependence of the assumptions of the theorem on u0subscript𝑢0u_{0} should be made explicit, because this theorem is applied for u0subscript𝑢0u_{0} replaced by a sequence of u0,n0subscript𝑢0𝑛superscript0u_{0,n}\to 0^{-}. So we compare it with Theorem 1.2, which is already stated in its sharp form. First, when δ𝛿\delta is sufficiently small,

infϑS20δ|u0|2(|Ωχ^|2+2|Lϕ|2)(u¯,u0,ϑ)du¯4Ω02(u0)(m2m1)subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0𝛿superscriptsubscript𝑢02superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢0italic-ϑdifferential-dsuperscript¯𝑢4superscriptsubscriptΩ02subscript𝑢0subscript𝑚2subscript𝑚1\displaystyle\inf_{\vartheta\in S^{2}}\int_{0}^{\delta}|u_{0}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u}^{\prime},u_{0},\vartheta)\mathrm{d}\underline{u}^{\prime}\approx 4\Omega_{0}^{2}(u_{0})(m_{2}-m_{1})

where misubscript𝑚𝑖m_{i} is the Hawking mass of Sisubscript𝑆𝑖S_{i}. Second, we have the equation

Dr=r2Ωtrχ¯𝐷𝑟𝑟2¯Ωtr𝜒\displaystyle Dr=\frac{r}{2}\overline{\Omega\mathrm{tr}\chi}

where Ωtrχ¯¯Ωtr𝜒\overline{\Omega\mathrm{tr}\chi} is the average of ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi in Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u}. Recalling r2Ωtrχ𝑟2Ωtr𝜒\frac{r}{2}\Omega\mathrm{tr}\chi takes value Ω02hsuperscriptsubscriptΩ02\Omega_{0}^{2}h on C¯0subscript¯𝐶0\underline{C}_{0}, then

r2r1Ω02(u0)h(u0)δ,subscript𝑟2subscript𝑟1superscriptsubscriptΩ02subscript𝑢0subscript𝑢0𝛿\displaystyle r_{2}-r_{1}\approx\Omega_{0}^{2}(u_{0})h(u_{0})\delta,

where risubscript𝑟𝑖r_{i} is the area radius of Sisubscript𝑆𝑖S_{i}. Therefore the lower bound conditions (1.1) and (1.6) only differ by a factor h(u0)subscript𝑢0h(u_{0}). This difference is acceptable. At last, r2r1subscript𝑟2subscript𝑟1r_{2}-r_{1} (or Ω02(u0)δsuperscriptsubscriptΩ02subscript𝑢0𝛿\Omega_{0}^{2}(u_{0})\delta by the above analysis) in the logarithm factor in (1.1) is replaced in (1.6) by |u1|subscript𝑢1|u_{1}|, which is the place where a closed trapped surface will form as predicted in Theorem 1.6.

A simple discussion may help to understand why a closed trapped surface will form at u=u1𝑢subscript𝑢1u=u_{1} where u1subscript𝑢1u_{1} is defined by (1.5). Assume the regular solution exists to u=u1𝑢subscript𝑢1u=u_{1} and some correct a priori estimates are obtained. From the Raychaudhuri equation along outgoing null direction, Dtrχ=12(trχ)2|χ^|22|L^ϕ|2,𝐷trsuperscript𝜒12superscripttr𝜒2superscript^𝜒22superscript^𝐿italic-ϕ2D\mathrm{tr}\chi^{\prime}=-\frac{1}{2}(\mathrm{tr}\chi)^{2}-|\widehat{\chi}|^{2}-2|\widehat{L}\phi|^{2}, where L^=Ω1L^𝐿superscriptΩ1𝐿\widehat{L}=\Omega^{-1}L, we have, on Cusubscript𝐶𝑢C_{u},

trχ2h|u|Ω02(u)0δ(|Ωχ^|2+2|Lϕ|2)du¯.trsuperscript𝜒2𝑢superscriptsubscriptΩ02𝑢superscriptsubscript0𝛿superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|}\approx-\Omega_{0}^{-2}(u)\int_{0}^{\delta}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}.

In order that to ensure that trχ<0trsuperscript𝜒0\mathrm{tr}\chi^{\prime}<0 at u¯=δ¯𝑢𝛿\underline{u}=\delta, u=u1𝑢subscript𝑢1u=u_{1}, we require

Ω02(u1)0δ|u0|2(|Ωχ^|2+2|Lϕ|2)(u¯,u0,ϑ)du¯superscriptsubscriptΩ02subscript𝑢1superscriptsubscript0𝛿superscriptsubscript𝑢02superscriptΩ^𝜒22superscript𝐿italic-ϕ2¯𝑢subscript𝑢0italic-ϑdifferential-d¯𝑢\displaystyle\Omega_{0}^{-2}(u_{1})\int_{0}^{\delta}|u_{0}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}
\displaystyle\approx Ω02(u1)0δ|u|2(|Ωχ^|2+2|Lϕ|2)(u¯,u1,θ)du¯>2|u1|.superscriptsubscriptΩ02subscript𝑢1superscriptsubscript0𝛿superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2¯𝑢subscript𝑢1𝜃differential-d¯𝑢2subscript𝑢1\displaystyle\Omega_{0}^{-2}(u_{1})\int_{0}^{\delta}|u|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u},u_{1},\theta)\mathrm{d}\underline{u}>2|u_{1}|.

So the definition (1.5) is obtained from the lower bound (1.6).

Remark 1.8.

We make an important remark that it is only the relation written in the vector field L𝐿L

Ωχ^|Cu1Ωχ^|Cu0,Lϕ|Cu1Lϕ|Cu0formulae-sequenceevaluated-atΩ^𝜒subscript𝐶subscript𝑢1evaluated-atΩ^𝜒subscript𝐶subscript𝑢0evaluated-at𝐿italic-ϕsubscript𝐶subscript𝑢1evaluated-at𝐿italic-ϕsubscript𝐶subscript𝑢0\displaystyle\Omega\widehat{\chi}\big{|}_{C_{u_{1}}}\approx\Omega\widehat{\chi}\big{|}_{C_{u_{0}}},\ L\phi\big{|}_{C_{u_{1}}}\approx L\phi\big{|}_{C_{u_{0}}}

can be proven but not the relation written in L^=Ω1L^𝐿superscriptΩ1𝐿\widehat{L}=\Omega^{-1}L. This is because only written in L𝐿L the equations for D¯(Ωχ^)¯𝐷Ω^𝜒\underline{D}(\Omega\widehat{\chi}) and D¯Lϕ¯𝐷𝐿italic-ϕ\underline{D}L\phi do not involve ω¯¯𝜔\underline{\omega}, whose initial value on C¯0subscript¯𝐶0\underline{C}_{0} has no a priori bounds.

The investigation of the problem of the formation of trapped surfaces when no symmetries are imposed was started in the celebrated work of Christodoulou [9], which also opened the path to the study of large data problem without symmetries. His theorem can be stated in a form similar to Theorem 1.6 as follows.

Theorem 1.7 (Christodoulou, [9]).

Suppose that the initial incoming null cone C¯0subscript¯𝐶0\underline{C}_{0} is a null cone in Minkowski space, and the smooth initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} satisfies

|u0|sup0u¯δi=03δiDiχ^5(u¯,u0)δ12Fsubscript𝑢0subscriptsupremum0¯𝑢𝛿superscriptsubscript𝑖03superscript𝛿𝑖subscriptnormsuperscript𝐷𝑖^𝜒superscript5¯𝑢subscript𝑢0superscript𝛿12𝐹\displaystyle|u_{0}|\sup_{0\leq\underline{u}\leq\delta}\sum_{i=0}^{3}\delta^{i}\|D^{i}\widehat{\chi}\|_{\mathbb{H}^{5}(\underline{u},u_{0})}\leq\delta^{-\frac{1}{2}}F

for some constants δ𝛿\delta, F𝐹F, and Ω1Ω1\Omega\equiv 1 on Cu0C¯0subscript𝐶subscript𝑢0subscript¯𝐶0C_{u_{0}}\bigcup\underline{C}_{0}. Then there exists a function M𝑀M of two variable such that if δM(F,u1)𝛿𝑀𝐹subscript𝑢1\delta\leq M(F,u_{1}), the smooth solution of the vacuum Einstein equations exists for 0u¯δ0¯𝑢𝛿0\leq\underline{u}\leq\delta, u0uu1subscript𝑢0𝑢subscript𝑢1u_{0}\leq u\leq u_{1}. If also

infϑS20δ|u0|2|χ^|2(u¯,u0,ϑ)du¯>2|u1|,subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0𝛿superscriptsubscript𝑢02superscript^𝜒2¯𝑢subscript𝑢0italic-ϑdifferential-d¯𝑢2subscript𝑢1\displaystyle\inf_{\vartheta\in S^{2}}\int_{0}^{\delta}|u_{0}|^{2}|\widehat{\chi}|^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}>2|u_{1}|, (1.7)

then Sδ,u1subscript𝑆𝛿subscript𝑢1S_{\delta,u_{1}} is a closed trapped surface after modifying M𝑀M to a smaller function M~~𝑀\widetilde{M}.

Remark 1.9.

Christodoulou only proved this theorem when u1=1subscript𝑢11u_{1}=-1. But it is trivial to extend the proof when u1[u0,0)subscript𝑢1subscript𝑢00u_{1}\in[u_{0},0) is arbitrary. In addition, Christodoulou constructed a class of initial data satisfying the assumptions of this theorem in his short pulse ansatz. He confirmed that the closed trapped surfaces are evolutionary.

Remark 1.10.

The choice of δ𝛿\delta in the above theorem is in particular independent of u0subscript𝑢0u_{0} when |u0|1subscript𝑢01|u_{0}|\geq 1. This fact allowed Christodoulou to pull the initial data back to the past null infinity, and then the condition (1.7) has a clear physical meaning. In contrast, Theorem 1.6, which we prove in this paper, allows us to push the initial data deep into the vertex.

The main difference between Theorem 1.7 and Theorem 1.6 is the the difference between two lower bound conditions (1.7) and (1.6). When δ𝛿\delta is suffciently small, the lower bound in (1.7) is much large than (1.6), and we have the following consequences: compared to the closed trapped surface forming in Theorem 1.7, the closed trapped surface forming in Theorem 1.6 is much smaller, and is located much closer to the vertex. Therefore, more refined a priori estimates are need to capture the growth properties of the solution approaching the vertex in Theorem 1.6. The first extension of Theorem 1.7 by relaxing the lower bound condition (1.7) to a condition similar to (1.6) was given by An-Luk [2]. They proved

Theorem 1.8 (An-Luk, [2]).

Suppose that the initial incoming null cone C¯0subscript¯𝐶0\underline{C}_{0} is a null cone in Minkowski space, and the smooth initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} with u0=1subscript𝑢01u_{0}=-1 satisfies

sup0u¯δχ^H7(u¯,u0)A12subscriptsupremum0¯𝑢𝛿subscriptnorm^𝜒superscript𝐻7¯𝑢subscript𝑢0superscript𝐴12\displaystyle\sup_{0\leq\underline{u}\leq\delta}\|\widehat{\chi}\|_{H^{7}(\underline{u},u_{0})}\leq A^{\frac{1}{2}}

and Ω1Ω1\Omega\equiv 1 on Cu0C¯0subscript𝐶subscript𝑢0subscript¯𝐶0C_{u_{0}}\bigcup\underline{C}_{0}. Then there exists a universal large constant b0subscript𝑏0b_{0} such that if b0bAsubscript𝑏0𝑏𝐴b_{0}\leq b\leq A and δA12b<1𝛿superscript𝐴12𝑏1\delta A^{\frac{1}{2}}b<1, then the smooth solution to the vacuum Einstein equations exists for 0u¯δ,1uδA12bformulae-sequence0¯𝑢𝛿1𝑢𝛿superscript𝐴12𝑏0\leq\underline{u}\leq\delta,-1\leq u\leq-\delta A^{\frac{1}{2}}b. Moreover, if the initial data also verify the lower bound

infϑ0δ|χ^|2(u¯,1,ϑ)du¯4bδA12,subscriptinfimumitalic-ϑsuperscriptsubscript0𝛿superscript^𝜒2¯𝑢1italic-ϑdifferential-d¯𝑢4𝑏𝛿superscript𝐴12\displaystyle\inf_{\vartheta}\int_{0}^{\delta}|\widehat{\chi}|^{2}(\underline{u},-1,\vartheta)\mathrm{d}\underline{u}\geq 4b\delta A^{\frac{1}{2}}, (1.8)

then Sδ,bδA12subscript𝑆𝛿𝑏𝛿superscript𝐴12S_{\delta,-b\delta A^{\frac{1}{2}}} is a closed trapped surface.

Setting ϕ0italic-ϕ0\phi\equiv 0, Theorem 1.6 recovers a special case of the above theorem if we set A12=a|log|u1||u0||superscript𝐴12𝑎subscript𝑢1subscript𝑢0A^{\frac{1}{2}}=a\left|\log\frac{|u_{1}|}{|u_{0}|}\right|, b0=C12subscript𝑏0superscriptsubscript𝐶12b_{0}=C_{1}^{2} and b=C2𝑏superscript𝐶2b=C^{2}. A similar condition to bA𝑏𝐴b\leq A is not required in Theorem 1.6 but holds when u1subscript𝑢1u_{1} is chosen sufficiently close to zero. However, the lower bound condition (1.8) is even better than (1.6), since we can choose A𝐴A to be a constant in Theorem 1.8. We should remark that the more general Theorem 5.1 can completely recover Theorem 1.8 in vacuum. Nevertheless, when the scalar field is presented, it is not obvious that we can choose A𝐴A, or the corresponding quantity in Theorem 5.1 to be a constant in general situations. There are at least two reasons for this. The first reason is due to a loss on the a priori estimates, which we call \mathscr{E}, a quantity involving derivatives of Lϕ𝐿italic-ϕL\phi and growing like |log|u1||u0||12superscriptsubscript𝑢1subscript𝑢012\left|\log\frac{|u_{1}|}{|u_{0}|}\right|^{\frac{1}{2}}. We will explain this quantity in Section 1.6. This quantity prevents us from solving the solution even deeper into the future than u=u1𝑢subscript𝑢1u=u_{1} defined in (1.5). Another reason is that φ=|u|Lϕ|C¯0𝜑evaluated-at𝑢𝐿italic-ϕsubscript¯𝐶0\varphi=|u|L\phi|_{\underline{C}_{0}} may be unbounded, which prevents the energy integral

0δ|u|2|Lϕ|2(u¯,u,ϑ)du¯superscriptsubscript0𝛿superscript𝑢2superscript𝐿italic-ϕ2¯𝑢𝑢italic-ϑdifferential-d¯𝑢\int_{0}^{\delta}|u|^{2}|L\phi|^{2}(\underline{u},u,\vartheta)\mathrm{d}\underline{u}

from being lower bounded before a closed trapped surface forms eventually. On the other hand, both the quantity \mathscr{E} and the estimate (1.4) suggest that the lower bound (1.6) is sharp up to the factor h(u0)subscript𝑢0h(u_{0}) in general.

There are also many other extensions to Theorem 1.7. The first extension and simplification was given by Klainerman-Rodnianski [14], by proving the geometric quantities obey some scale-invariant estimates relative to the parabolic scaling on null hypersurfaces. Although they only considered a finite problem, An [1] extended it to an initial value problem on the past null infinity. Luk-Rodnianski proved in [22] among many other things that the formation of trapped surfaces theorem also holds when C¯0subscript¯𝐶0\underline{C}_{0} is not necessarily Minkowskian, while the other existed works should assume C¯0subscript¯𝐶0\underline{C}_{0} to be Minkowskian. Klainerman-Luk-Rodnianski proved in [13] the lower bound condition (1.7) can be relaxed, by replacing infinfimum\inf by supsupremum\sup. Similar extensions are expected to hold for Theorems 1.6 and 1.8. We also point out the first paper on the formation of trapped surfaces for non-vacuum equations by Yu [24], which considered the Einstein equations coupled with the electric-magnetic field, and a paper by the first author and Yu [19], in which a class of asymptotically flat Cauchy data whose future development has a closed trapped surface was constructed, while the other existed works were about characteristic initial value problem.

1.5. Instability theorems

We will discuss the instability theorems in this subsection. The instability theorem proved in [8], Theorem 1.3 should also be generalized beyond spherical symmetry. The original proof of Theorem 1.3 is by contradiction. By introducing the dimensionless coordinates (t,s)𝑡𝑠(t,s) instead of (u,r)𝑢𝑟(u,r), the Bondi coordinate, by

u=u0et,2r=u0est,formulae-sequence𝑢subscript𝑢0superscripte𝑡2𝑟subscript𝑢0superscripte𝑠𝑡\displaystyle u=u_{0}\mathrm{e}^{-t},\ -2r=u_{0}\mathrm{e}^{s-t},

that the conclusion of Theorem 1.1 is false implies that there exists some ε>0𝜀0\varepsilon>0 such that

2(m(s,t)m(0,t))c1r(s,t)slog(1s),2𝑚𝑠𝑡𝑚0𝑡subscript𝑐1𝑟𝑠𝑡𝑠1𝑠\displaystyle 2(m(s,t)-m(0,t))\leq c_{1}r(s,t)s\log\left(\frac{1}{s}\right), (1.9)

for all (s,t){sc0}{0u¯ε}𝑠𝑡𝑠subscript𝑐00¯𝑢𝜀(s,t)\in\{s\leq c_{0}\}\bigcap\{0\leq\underline{u}\leq\varepsilon\}. Then (1.9) is used to conclude that the opposite of (1.9) , or equivalently (1.1) holds, however for some (sε,tε){sc0}{0u¯ε}subscript𝑠𝜀subscript𝑡𝜀𝑠subscript𝑐00¯𝑢𝜀(s_{\varepsilon},t_{\varepsilon})\in\{s\leq c_{0}\}\bigcap\{0\leq\underline{u}\leq\varepsilon\}, allowing a perturbation on the initial data. This argument also implicitly use the fact that the solution exists in the region {sc0}{0u¯ε}𝑠subscript𝑐00¯𝑢𝜀\{s\leq c_{0}\}\bigcap\{0\leq\underline{u}\leq\varepsilon\}.

If we go beyond spherical symmetry, the above arguments may break down for at least two reasons. First, a similar assumption to (1.9) is far from enough to control the whole system of the Einstein equations, in contrast to the spherically symmetric case, where the Hawking mass m𝑚m governs the whole system. Second, the existence region of the solution relies on some suitable a priori estimates of the solution, which is also different from the spherically symmetric case. For these two reasons, instead of (1.9), an appropriate a priori estimate is needed, to guarantee that the solution exists deep enough into the future such that the assumptions of Theorem 1.6 eventually hold.

Remark 1.11.

It is not difficult to imagine that one should first find another robust argument of proving Theorem 1.1, or more precisely, Theorem 1.3, and then generalize it to the problem we study in this paper. We are able to find this argument and this is given in detail in a separated paper [16].

Refer to caption
Figure 2. The conclusions of Theorem 1.9 and 1.10

The instability theorems we prove in this paper is Theorem LABEL:instabilitytheorem and Corollary LABEL:instabilitycorollary. We state a rough version here and depict them in Figure 2. Recall that the proof of Theorem 1.3 in [8] is divided into several cases according to the asymptotic behavior of φ𝜑\varphi as u0𝑢superscript0u\to 0^{-}. Similarly, we divide the proof of the instability theorems in two separated cases in this paper. The first case is the following.

Theorem 1.9 (Case 1 of Theorem LABEL:instabilitytheorem).

If φ(u)𝜑𝑢\varphi(u) is unbounded as u0𝑢superscript0u\to 0^{-} and the data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} is smooth, then there exists two sequences δn0+subscript𝛿𝑛superscript0\delta_{n}\to 0^{+}, u0,n0subscript𝑢0𝑛superscript0u_{0,n}\to 0^{-} such that the smooth solution of the Einstein-scalar field equaitons exists for 0u¯δn0¯𝑢subscript𝛿𝑛0\leq\underline{u}\leq\delta_{n}, u0uu0,nsubscript𝑢0𝑢subscript𝑢0𝑛u_{0}\leq u\leq u_{0,n} and the assumptions of Theorem 1.6 hold for δ=δn𝛿subscript𝛿𝑛\delta=\delta_{n}, u0=u0,nsubscript𝑢0subscript𝑢0𝑛u_{0}=u_{0,n} for all n𝑛n. Consequently, there exists a sequence u1,n0subscript𝑢1𝑛superscript0u_{1,n}\to 0^{-} such that the solution remains smooth for 0u¯δn0¯𝑢subscript𝛿𝑛0\leq\underline{u}\leq\delta_{n}, u0,nu0u1,nsubscript𝑢0𝑛subscript𝑢0subscript𝑢1𝑛u_{0,n}\leq u_{0}\leq u_{1,n} and Sδn,u1,nsubscript𝑆subscript𝛿𝑛subscript𝑢1𝑛S_{\delta_{n},u_{1,n}} are closed trapped surfaces for all n𝑛n.

In the first case, nothing essential are required on the initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}}. In the second case when φ(u)𝜑𝑢\varphi(u) is bounded, it is more complicated. We introduce a monotonically decreasing positive function δ~=δ~(u~)~𝛿~𝛿~𝑢\widetilde{\delta}=\widetilde{\delta}(\widetilde{u}) defined for u~[u0,0)~𝑢subscript𝑢00\widetilde{u}\in[u_{0},0) where δ~0+~𝛿superscript0\widetilde{\delta}\to 0^{+} as u~0~𝑢superscript0\widetilde{u}\to 0^{-}. A crucial fact is that the definition of this function is independent of the initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}}. We will not write down the explicit expression here but it is designed based on the following idea: it is expected that the assumptions of Theorem 1.6 hold for δ=δ~𝛿~𝛿\delta=\widetilde{\delta}, u0=u~subscript𝑢0~𝑢u_{0}=\widetilde{u}. Now assume we have such a function δ~=δ~(u~)~𝛿~𝛿~𝑢\widetilde{\delta}=\widetilde{\delta}(\widetilde{u}). We define another function

f(u~)=1δ~(u~)infϑS20δ~(u~)(|u0|2|Ωχ^|2(u¯,u0,ϑ)+||u0|Lϕ(u¯,u0,ϑ)+(φ(u~)φ(u0))|2)du¯.𝑓~𝑢1~𝛿~𝑢subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0~𝛿~𝑢superscriptsubscript𝑢02superscriptΩ^𝜒2¯𝑢subscript𝑢0italic-ϑsuperscriptsubscript𝑢0𝐿italic-ϕ¯𝑢subscript𝑢0italic-ϑ𝜑~𝑢𝜑subscript𝑢02differential-d¯𝑢\displaystyle f(\widetilde{u})=\frac{1}{\widetilde{\delta}(\widetilde{u})}\inf_{\vartheta\in S^{2}}\int_{0}^{\widetilde{\delta}(\widetilde{u})}(|u_{0}|^{2}|\Omega\widehat{\chi}|^{2}(\underline{u},u_{0},\vartheta)+||u_{0}|L\phi(\underline{u},u_{0},\vartheta)+(\varphi(\widetilde{u})-\varphi(u_{0}))|^{2})\mathrm{d}\underline{u}. (1.10)

Then we have the following theorem about the second case.

Theorem 1.10 (Case 2 of Theorem LABEL:instabilitytheorem and Corollary LABEL:instabilitycorollary).

Suppose that φ(u)𝜑𝑢\varphi(u) is bounded and the data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} is smooth. We also fix some γ(0,2)𝛾02\gamma\in(0,2) on which the definition of the function δ~=δ~(u~)~𝛿~𝛿~𝑢\widetilde{\delta}=\widetilde{\delta}(\widetilde{u}) also depends. Then there exists some ε>0𝜀0\varepsilon>0 such that if we can find some |u~|<ε~𝑢𝜀|\widetilde{u}|<\varepsilon such that

Ω0γ2(u~)f(u~)2,superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢2\displaystyle\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u})\geq 2,

then the smooth solution of the Einstein-scalar field equations exists for 0δδ~0𝛿~𝛿0\leq\delta\leq\widetilde{\delta}, u0uu~subscript𝑢0𝑢~𝑢u_{0}\leq u\leq\widetilde{u} and the assumptions of Theorem 1.6 hold for δ=δ~𝛿~𝛿\delta=\widetilde{\delta}, u0=u~subscript𝑢0~𝑢u_{0}=\widetilde{u}. Consequently, there exists some u~subscript~𝑢\widetilde{u}_{*} such that the solution remains smooth for 0u¯δ~0¯𝑢~𝛿0\leq\underline{u}\leq\widetilde{\delta}, u~uu~~𝑢𝑢subscript~𝑢\widetilde{u}\leq u\leq\widetilde{u}_{*} and Sδ~,u~subscript𝑆~𝛿subscript~𝑢S_{\widetilde{\delta},\widetilde{u}_{*}} is a closed trapped surface. If in addition

lim supu~0Ω0γ2(u~)f(u~)>2,subscriptlimit-supremum~𝑢superscript0superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢2\displaystyle\limsup_{\widetilde{u}\to 0^{-}}\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u})>2, (1.11)

then the above conclusions hold for three sequences δ~=δ~n0+~𝛿subscript~𝛿𝑛superscript0\widetilde{\delta}=\widetilde{\delta}_{n}\to 0^{+}, u~=u~0,n0~𝑢subscript~𝑢0𝑛superscript0\widetilde{u}=\widetilde{u}_{0,n}\to 0^{-}, u~=u~1,n0subscript~𝑢subscript~𝑢1𝑛superscript0\widetilde{u}_{*}=\widetilde{u}_{1,n}\to 0^{-}.

Remark 1.12.

In Remark 1.3, we have mentioned that fully anisotropic instability theorems are needed to prove the positive co-dimensionality of the exceptional set in \mathcal{I}. This is to say we will try to replace the infinfimum\inf in the definition (1.10) by a supsupremum\sup in future works. This can be achieved by proving a correct anisotropic version of Theorem 1.6 based on the techniques introduced in the work [13].

Note that in general we cannot choose \mathcal{I} to be the space of all smooth initial conformal metrics from the form of condition (1.11). Therefore we should study a more general class of the initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} including non-smooth data. For the Cauchy problem of the Einstein equations with non-smooth initial data, a best result is a recent development which is the resolution of the bounded L2superscript𝐿2L^{2} conjecture by Klainerman-Rodnianski-Szeftel (see [15]), allowing the curvature of the initial metric lying in L2superscript𝐿2L^{2}. For the characteristic problem, the classical theory (see [23]) only provides us a way to solve the equation with sufficiently smooth initial data. Recently, Luk-Rodnianski (see [21]) developed a theory to solve by limiting argument the Einstein equations with initial α𝛼\alpha not even in L2superscript𝐿2L^{2} relative to u¯¯𝑢\underline{u} (but the other curvature components should have more regularity). A key ingredient is, they developed a renormalization technique of the Bianchi equations such that the a priori estimates can be derived for initial shear χ^^𝜒\widehat{\chi} only lying in Lsuperscript𝐿L^{\infty} relative to u¯¯𝑢\underline{u} (but χ^^𝜒\widehat{\chi} should have several order of the angular derivatives) and the convergence and uniqueness can also be proved. In this paper, we also choose to work in the space of initial data such that χ^L^𝜒superscript𝐿\widehat{\chi}\in L^{\infty} relative to u¯¯𝑢\underline{u}. It is natural in view of the work [21, 2], since the a priori estimates are done in terms of that norm. Moreover, we require in addition χ^C0^𝜒superscript𝐶0\widehat{\chi}\in C^{0} relative to u¯¯𝑢\underline{u} up to the intersection Cu0C¯0subscript𝐶subscript𝑢0subscript¯𝐶0C_{u_{0}}\cap\underline{C}_{0}, which is the completion of Csuperscript𝐶C^{\infty} in the norm Lsuperscript𝐿L^{\infty}. It is better to prove the genericity theorems in more regular space because this would mean the singularity is more unstable. But Lsuperscript𝐿L^{\infty} relative to u¯¯𝑢\underline{u}, or C0superscript𝐶0C^{0}, is the strongest norm on which the a priori estimates are based, and in which we can prove the genericity theorems using the techniques in this paper.

In principle, one should develop a theory to solve the equations with non-smooth initial data, or apply the theory developed in [21] with some modifications. However, in order not to obscure our main idea, we are not going to write down the full detail of developing such a theory. Instead, we state and prove Theorem 6.2, which can be stated roughly as follows.

Theorem 1.11.

Fix some singular initial data on C¯0subscript¯𝐶0\underline{C}_{0} as above. Suppose that the initial data (g/^,Ω,ϕ)(\widehat{\mbox{$g\mkern-9.0mu/$}},\Omega,\phi) on Cu0subscript𝐶subscript𝑢0C_{u_{0}} satisfies χ^,ω,LϕLu¯HN(Su¯,u0)^𝜒𝜔𝐿italic-ϕsubscriptsuperscript𝐿¯𝑢superscript𝐻𝑁subscript𝑆¯𝑢subscript𝑢0\widehat{\chi},\omega,L\phi\in L^{\infty}_{\underline{u}}H^{N}(S_{\underline{u},u_{0}}) and LϕLϕ¯Hu¯1HN(Su¯,u0)𝐿italic-ϕ¯𝐿italic-ϕsubscriptsuperscript𝐻1¯𝑢superscript𝐻𝑁subscript𝑆¯𝑢subscript𝑢0L\phi-\overline{L\phi}\in H^{1}_{\underline{u}}H^{N}(S_{\underline{u},u_{0}}) for sufficiently large integer N𝑁N. If φ(u)𝜑𝑢\varphi(u) is bounded, we assume in addition

lim supu~0Ω0γ2(u~)f(u~)>32.subscriptlimit-supremum~𝑢superscript0superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢32\displaystyle\limsup_{\widetilde{u}\to 0^{-}}\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u})>32. (1.12)

Suppose also that we have a sequence of smooth initial data (g/^n,Ωn,ϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},\phi_{n}) such that χ^nsubscript^𝜒𝑛\widehat{\chi}_{n} and (Lϕ)nsubscript𝐿italic-ϕ𝑛(L\phi)_{n} converge to χ^^𝜒\widehat{\chi} and Lϕ𝐿italic-ϕL\phi in LϑLu¯2subscriptsuperscript𝐿italic-ϑsubscriptsuperscript𝐿2¯𝑢L^{\infty}_{\vartheta}L^{2}_{\underline{u}}. Then there exist two sequences δk0+subscript𝛿𝑘superscript0\delta_{k}\to 0^{+}, u1,k0subscript𝑢1𝑘superscript0u_{1,k}\to 0^{-} such that, for every k𝑘k, there exists some Nksubscript𝑁𝑘N_{k} such that for all n>Nk𝑛subscript𝑁𝑘n>N_{k}, Sδk,u1,ksubscript𝑆subscript𝛿𝑘subscript𝑢1𝑘S_{\delta_{k},u_{1,k}} is a closed strictly trapped surface in the maximal future development of (g/^n,Ωn,ϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},\phi_{n}).

Here the strictness means that both null expansions of Sδk,u1,ksubscript𝑆subscript𝛿𝑘subscript𝑢1𝑘S_{\delta_{k},u_{1,k}} in the maximal future development of (g/^n,Ωn,ϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},\phi_{n}) are less than a negative number, which depends on k𝑘k but is independent of n>Nk𝑛subscript𝑁𝑘n>N_{k}. This theorem has the following implication: If the corresponding sequence of smooth solutions has a limiting spacetime such that both outgoing and incoming null expansions converge pointwisely, then the limiting spacetime has a sequence of closed trapped surfaces approaching the singularity. The limiting spacetime is clearly not unique, and even not necessarily satisfies the Einstein equations. However, any theories of solving the Einstein-scalar field equations with non-smooth initial data by limiting argument, satisfying the requirements in Theorem 1.11 are included. Therefore, we may say that the maximal future development of (g/^,Ω,ϕ)(\widehat{\mbox{$g\mkern-9.0mu/$}},\Omega,\phi) satisfying the assumptions of Theorem 1.11 has a sequence of closed trapped surfaces approaching the singularity. Then finally, the proof of Theorem 1.4 is to show that

{g/^|lim supu~0Ω0γ2(u~)f(u~)>32}\displaystyle\{\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}|\limsup_{\widetilde{u}\to 0^{-}}\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u})>32\}\subset\mathcal{E}

contains a countably intersection of open and dense subsets of \mathcal{I}. This can be achieved by showing that

{g/^| there exists some u~ with δ~(u~)<ε such that Ω0γ2(u~)f(u~)>33}\displaystyle\{\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}|\text{ there exists some $\widetilde{u}$ with $\widetilde{\delta}(\widetilde{u})<\varepsilon$ such that }\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u})>33\}

is open and dense for all ε>0𝜀0\varepsilon>0. Indeed, the above set is a subset of εsubscript𝜀\mathcal{E}_{\varepsilon} for sufficiently ε>0𝜀0\varepsilon>0 and this also proves Theorem 1.5 in view of Theorem 1.10.

1.6. A priori estimates and the existence theorem

To establish the formation of trapped surface theorem and the instability theorems, the most important part is prove the existence of the solution. For the initial value problem of the Einstein equations without symmetries, this is usually the most lengthy and technical part. The basic strategy is to do a priori estimates with suitable weight functions through L2superscript𝐿2L^{2} based energy method, which can be traced back to the celebrated work [10] of the stability of Minkowski space by Christodoulou and Klainerman. While the stability of Minkowski space is essentially a small data problem, the study of large data problem was pioneered by Christodoulou in the work [9] on the formation of trapped surface.

There are statements on the existence of the solutions in Theorem 1.6, 1.9 and 1.10. We need to prove two parts of the existence. The first part is to solve the solution from u=u0𝑢subscript𝑢0u=u_{0} to u=u~𝑢~𝑢u=\widetilde{u} where the assumptions of Theorem 1.6 hold, which is included in the statement of Theorem 1.9 and 1.10. The second part is to solve the solution from u=u~𝑢~𝑢u=\widetilde{u} to u=u~𝑢subscript~𝑢u=\widetilde{u}_{*} where a closed trapped surface will form, which is included in the statement of Theorem 1.6. It turns out that these two parts of the existence can be included in a general form of existence theorem, which is Theorem 3.1 in this paper. Let us state it without a statement on the quantitative behavior of the solution as follows.

Theorem 1.12.

There exists a universal constant C01subscript𝐶01C_{0}\geq 1 such that the following statement is true. Let C𝐶C, u0subscript𝑢0u_{0} and u1subscript𝑢1u_{1} be three number such that CC0𝐶subscript𝐶0C\geq C_{0} and u0<u1<0subscript𝑢0subscript𝑢10u_{0}<u_{1}<0. The smooth initial data on C¯0subscript¯𝐶0\underline{C}_{0} is spherically symmetric (not necessarily singular) and on Cu0subscript𝐶subscript𝑢0C_{u_{0}}, satisfies

𝒜=𝒜(δ,u0,u1):=max{1,supu0uu1(1|φ(u)|),1|u0|sup0u¯δ(Ωχ^7(u¯,u0)+ω5(u¯,u0)+Lϕ5(u¯,u0))}<+.𝒜𝒜𝛿subscript𝑢0subscript𝑢1assign1subscriptsupremumsubscript𝑢0𝑢subscript𝑢1superscript1𝜑𝑢superscript1subscript𝑢0subscriptsupremum0¯𝑢𝛿subscriptdelimited-∥∥Ω^𝜒superscript7¯𝑢subscript𝑢0subscriptdelimited-∥∥𝜔superscript5¯𝑢subscript𝑢0subscriptdelimited-∥∥𝐿italic-ϕsuperscript5¯𝑢subscript𝑢0\begin{split}\mathcal{A}=\mathcal{A}(\delta,u_{0},u_{1}):=&\max\left\{1,\sup_{u_{0}\leq u\leq u_{1}}\left(\mathscr{F}^{-1}|\varphi(u)|\right),\right.\\ &\left.\mathscr{F}^{-1}|u_{0}|\sup_{0\leq\underline{u}\leq\delta}\left(\|\Omega\widehat{\chi}\|_{\mathbb{H}^{7}(\underline{u},u_{0})}+\|\omega\|_{\mathbb{H}^{5}(\underline{u},u_{0})}+\|L\phi\|_{\mathbb{H}^{5}(\underline{u},u_{0})}\right)\right\}<+\infty.\end{split} (1.13)

for some function =(δ,u0,u1)1𝛿subscript𝑢0subscript𝑢11\mathscr{F}=\mathscr{F}(\delta,u_{0},u_{1})\geq 1, and Ω0(u0)1subscriptΩ0subscript𝑢01\Omega_{0}(u_{0})\leq 1. Denote

=(δ,u0,u1):=𝛿subscript𝑢0subscript𝑢1assignabsent\displaystyle\mathscr{E}=\mathscr{E}(\delta,u_{0},u_{1}):= max{1,1𝒜1|u0|(|u0|/ )Lϕ𝕃[0,δ]24(u0)|log|u1||u0||12}1,1superscript1superscript𝒜1evaluated-atsubscript𝑢0subscript𝑢0/ 𝐿italic-ϕsubscriptsuperscript𝕃20𝛿superscript4subscript𝑢0superscriptsubscript𝑢1subscript𝑢0121\displaystyle\max\left\{1,\mathscr{F}^{-1}\mathcal{A}^{-1}\||u_{0}|(|u_{0}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{[0,\delta]}\mathbb{H}^{4}(u_{0})}\left|\log\frac{|u_{1}|}{|u_{0}|}\right|^{\frac{1}{2}}\right\}\geq 1, (1.14)
𝒲=𝒲(u0,u1):=𝒲𝒲subscript𝑢0subscript𝑢1assignabsent\displaystyle\mathscr{W}=\mathscr{W}(u_{0},u_{1}):= max{1,|logΩ0(u1)Ω0(u0)|}1.1subscriptΩ0subscript𝑢1subscriptΩ0subscript𝑢01\displaystyle\max\left\{1,\left|\log\frac{\Omega_{0}(u_{1})}{\Omega_{0}(u_{0})}\right|\right\}\geq 1. (1.15)

Suppose that the following smallness conditions hold:

Ω02(u0)δ|u1|12𝒲superscriptsubscriptΩ02subscript𝑢0𝛿superscriptsubscript𝑢11superscript2𝒲\displaystyle\Omega_{0}^{2}(u_{0})\delta|u_{1}|^{-1}\mathscr{E}^{2}\mathscr{W} 1,less-than-or-similar-toabsent1\displaystyle\lesssim 1, (1.16)
C2δ|u1|1𝒲𝒜superscript𝐶2𝛿superscriptsubscript𝑢11𝒲𝒜\displaystyle C^{2}\delta|u_{1}|^{-1}\mathscr{F}\mathscr{W}\mathcal{A} 1,absent1\displaystyle\leq 1, (1.17)

and the following auxiliary condition holds:

Ω02(u0)Ω02(u1)δ|u1|1𝒜1.less-than-or-similar-tosuperscriptsubscriptΩ02subscript𝑢0superscriptsubscriptΩ02subscript𝑢1𝛿superscriptsubscript𝑢11𝒜1\begin{split}\Omega_{0}^{2}(u_{0})\Omega_{0}^{-2}(u_{1})\delta|u_{1}|^{-1}\mathscr{F}\mathcal{A}&\lesssim 1.\end{split} (1.18)

Then the smooth solution of the Einstein-scalar field equations exists in the region 0u¯δ0¯𝑢𝛿0\leq\underline{u}\leq\delta, u0uu1subscript𝑢0𝑢subscript𝑢1u_{0}\leq u\leq u_{1}.

The proof is by establishing the a priori estimates. One of the main difficulties of the proof is that we need to solve the equations deep near the singular vertex, approaching which the geometric quantities may growth. This has been handled first by [2], i.e., Theorem 1.8 where C¯0subscript¯𝐶0\underline{C}_{0} is assumed to be Minkowskian and the vertex is of course regular. On the other hand, the scalar field and the singularity bring us new difficulties. In the following, we will briefly describe the a priori estimates we want to prove. We will also explain the quantities and the conditions introduced in the above theorem. We will encounter several difficulties and mention how to overcome them.

1.6.1. Quantitative behavior of the connection coefficients and the scalar field

We begin by analyzing the expected behaviors of the connection coefficients and the scalar field, which are similar to those in [2]. We should keep in mind that we hope to solve the solution up to u=u1𝑢subscript𝑢1u=u_{1} for 0u¯δ0¯𝑢𝛿0\leq\underline{u}\leq\delta. By scaling consideration, we will expect that, for u¯[0,δ],u[u0,u1]formulae-sequence¯𝑢0𝛿𝑢subscript𝑢0subscript𝑢1\underline{u}\in[0,\delta],u\in[u_{0},u_{1}],

Θ|u|1𝒜whereΘ{Ωχ^,Ωtrχ,Lϕ}.less-than-or-similar-toΘsuperscript𝑢1𝒜whereΘΩ^𝜒Ωtr𝜒𝐿italic-ϕ\displaystyle\Theta\lesssim|u|^{-1}\mathscr{F}\mathcal{A}\ \text{where}\ \Theta\in\{\Omega\widehat{\chi},\Omega\mathrm{tr}\chi,L\phi\}. (1.19)

\mathscr{F} is chosen according to the problem at hand and 𝒜𝒜\mathcal{A} defined in (1.13) is the scale-invariant bound for the initial data. We usually choose supu0uu1(|φ(u)|)subscriptsupremumsubscript𝑢0𝑢subscript𝑢1𝜑𝑢\mathscr{F}\geq\displaystyle\sup_{u_{0}\leq u\leq u_{1}}\left(|\varphi(u)|\right) and hence 𝒜𝒜\mathscr{F}\mathcal{A} is essentially the scale-invariant bound for the initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}}. From the Raychaudhuri equation

D(trχ)=Ω2ΘΘ𝐷trsuperscript𝜒superscriptΩ2ΘΘ\displaystyle D(\mathrm{tr}\chi^{\prime})=\Omega^{-2}\Theta\cdot\Theta

and from that ΩΩ0ΩsubscriptΩ0\Omega\approx\Omega_{0} which is reasonable, ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi has another, in fact better estimate:

Ω02(trχ2|u|)δ|u|22𝒜2.less-than-or-similar-tosuperscriptsubscriptΩ02trsuperscript𝜒2𝑢𝛿superscript𝑢2superscript2superscript𝒜2\displaystyle\Omega_{0}^{2}(\mathrm{tr}\chi^{\prime}-\frac{2}{|u|})\lesssim\delta|u|^{-2}\mathscr{F}^{2}\mathcal{A}^{2}.

Now we turn to the following quantites

Θ/ {η,η¯,/ ϕ},Θ¯{Ωχ¯^,Ωtrχ¯+2|u|},L¯ϕψ|u|.formulae-sequenceΘ/ 𝜂¯𝜂/ italic-ϕ¯ΘΩ^¯𝜒Ωtr¯𝜒2𝑢¯𝐿italic-ϕ𝜓𝑢\displaystyle\mbox{$\Theta\mkern-12.0mu/$ }\in\{\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi\},\ \underline{\Theta}\in\{\Omega\widehat{\underline{\chi}},\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\},\ \underline{L}\phi-\frac{\psi}{|u|}.

They satisfy the following equations in a schematic form:

D(Ωχ¯^orΩtrχ¯+2|u|)=Ω2(/ Θ/ +Θ/ Θ/ )+Ωtrχ¯Θ+ΘΘ¯+curvature,D(L¯ϕψ|u|)=Ω2(/ Θ/ +Θ/ Θ/ )+[ΩtrχL¯ϕ]+Ωtrχ¯Θ,D(ηor/ ϕ)=/ Θ+ΘΘ/ ,D¯η¯=Θ¯Θ/ +Ωtrχ¯η+[L¯ϕ/ ϕ]+curvature.formulae-sequence𝐷Ω^¯𝜒orΩtr¯𝜒2𝑢superscriptΩ2/ Θ/ Θ/ Θ/ Ωtr¯𝜒ΘΘ¯Θcurvatureformulae-sequence𝐷¯𝐿italic-ϕ𝜓𝑢superscriptΩ2/ Θ/ Θ/ Θ/ delimited-[]Ωtr𝜒¯𝐿italic-ϕΩtr¯𝜒Θformulae-sequence𝐷𝜂or/ italic-ϕ/ ΘΘΘ/ ¯𝐷¯𝜂¯ΘΘ/ Ωtr¯𝜒𝜂delimited-[]¯𝐿italic-ϕ/ italic-ϕcurvature\begin{split}D(\Omega\widehat{\underline{\chi}}\ \text{or}\ \Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})&=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\mbox{$\Theta\mkern-12.0mu/$ }+\mbox{$\Theta\mkern-12.0mu/$ }\cdot\mbox{$\Theta\mkern-12.0mu/$ })+\text{\framebox{$\Omega\mathrm{tr}\underline{\chi}\Theta$}}+\Theta\cdot\underline{\Theta}+\text{curvature},\\ D(\underline{L}\phi-\frac{\psi}{|u|})&=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\mbox{$\Theta\mkern-12.0mu/$ }+\mbox{$\Theta\mkern-12.0mu/$ }\cdot\mbox{$\Theta\mkern-12.0mu/$ })+\Big{[}\Omega\mathrm{tr}\chi\underline{L}\phi\Big{]}+\text{\framebox{$\Omega\mathrm{tr}\underline{\chi}\Theta$}},\\ D(\eta\ \text{or}\ \mbox{$\nabla\mkern-13.0mu/$ }\phi)&=\text{\framebox{$\mbox{$\nabla\mkern-13.0mu/$ }\Theta$}}+\Theta\cdot\mbox{$\Theta\mkern-12.0mu/$ },\\ \underline{D}\underline{\eta}&=\underline{\Theta}\cdot\mbox{$\Theta\mkern-12.0mu/$ }+\text{\framebox{$\Omega\mathrm{tr}\underline{\chi}\eta$}}+\Big{[}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\Big{]}+\text{curvature}.\end{split} (1.20)

The boxed and square bracketed terms, which are the so-called borderline terms, determine the behavior of the corresponding quantities respectively. First of all, it is reasonable to expect that an angular derivative /\nabla\mkern-13.0mu/ costs an |u|1superscript𝑢1|u|^{-1}, and L¯ϕ¯𝐿italic-ϕ\underline{L}\phi, Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi}, which are non-vanishing on C¯0subscript¯𝐶0\underline{C}_{0}, remain close to their initial values:

L¯ϕψ|u|,Ωtrχ¯2|u|.formulae-sequence¯𝐿italic-ϕ𝜓𝑢Ωtr¯𝜒2𝑢\displaystyle\underline{L}\phi\approx\frac{\psi}{|u|},\ \Omega\mathrm{tr}\underline{\chi}\approx-\frac{2}{|u|}.

Then after integration, the first three boxed terms in (1.20) contribute an estimate like δ|u|2𝒜𝛿superscript𝑢2𝒜\delta|u|^{-2}\mathscr{F}\mathcal{A} and this suggests that

Θ/ ,Θ¯exceptη¯δ|u|2𝒜.less-than-or-similar-toΘ/ ¯Θexcept¯𝜂𝛿superscript𝑢2𝒜\displaystyle\mbox{$\Theta\mkern-12.0mu/$ },\underline{\Theta}\ \text{except}\ \underline{\eta}\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}. (1.21)

The last boxed term is then also contributes an estimate like δ|u|2𝒜𝛿superscript𝑢2𝒜\delta|u|^{-2}\mathscr{F}\mathcal{A} after integration. Now we should pay more attentions to the square bracketed terms, containing a factor L¯ϕ¯𝐿italic-ϕ\underline{L}\phi, which behaves only like its initial value, worse than that suggested by signature consideration666The signature is a number assigned to every geometric quantity according to how many e3=Ω1L¯subscript𝑒3superscriptΩ1¯𝐿e_{3}=\Omega^{-1}\underline{L} and how many e4=Ω1Lsubscript𝑒4superscriptΩ1𝐿e_{4}=\Omega^{-1}L are used in its definition. See for example [10] or [14] for a detailed description. Here L¯ϕ¯𝐿italic-ϕ\underline{L}\phi and Ωχ¯^Ω^¯𝜒\Omega\widehat{\underline{\chi}} have the same signature but they have totally different behaviors.. Moreover, its initial value ψ|u|𝜓𝑢\frac{\psi}{|u|} is not assumed to have any a priori bounds. The resolution of this difficulty is to use the equality (1.3) to estimate L¯ϕ¯𝐿italic-ϕ\underline{L}\phi in terms of the monotonically decreasing function Ω0subscriptΩ0\Omega_{0}, the restriction of ΩΩ\Omega on C¯0subscript¯𝐶0\underline{C}_{0}. The price is that Ω0subscriptΩ0\Omega_{0} then appears as a weight function in our estimates, and we will find out in the proof that this is not a big problem. Precisely, we will have

u0u|u||L¯ϕ|2du𝒲,where 𝒲=𝒲(u0,u1) is defined in (1.15).less-than-or-similar-tosuperscriptsubscriptsubscript𝑢0𝑢superscript𝑢superscript¯𝐿italic-ϕ2differential-dsuperscript𝑢𝒲where 𝒲=𝒲(u0,u1) is defined in (1.15)\displaystyle\int_{u_{0}}^{u}|u^{\prime}||\underline{L}\phi|^{2}\mathrm{d}u^{\prime}\lesssim\mathscr{W},\ \text{where $\mathscr{W}=\mathscr{W}(u_{0},u_{1})$ is defined in \eqref{def-W-intro}}.

Then the contribution from the square bracketed term of the last equation is

|u||η¯|+u0u|u||L¯ϕ||/ ϕ|duδ𝒜u0u|u|1|L¯ϕ|duδ|u|1𝒲12𝒜,less-than-or-similar-to𝑢¯𝜂superscriptsubscriptsubscript𝑢0𝑢superscript𝑢¯𝐿italic-ϕ/ italic-ϕdifferential-dsuperscript𝑢less-than-or-similar-to𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1¯𝐿italic-ϕdifferential-dsuperscript𝑢less-than-or-similar-to𝛿superscript𝑢1superscript𝒲12𝒜\displaystyle|u||\underline{\eta}|\lesssim\cdots+\int_{u_{0}}^{u}|u^{\prime}||\underline{L}\phi||\mbox{$\nabla\mkern-13.0mu/$ }\phi|\mathrm{d}u^{\prime}\lesssim\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-1}|\underline{L}\phi|\mathrm{d}u^{\prime}\lesssim\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A},

suggesting that

η¯δ|u|2𝒲12𝒜.less-than-or-similar-to¯𝜂𝛿superscript𝑢2superscript𝒲12𝒜\displaystyle\underline{\eta}\lesssim\delta|u|^{-2}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (1.22)

Finally we move to the second equation. By this equation, we have

L¯ϕψ|u|δ|u|2𝒜(|ψ|+1).less-than-or-similar-to¯𝐿italic-ϕ𝜓𝑢𝛿superscript𝑢2𝒜𝜓1\displaystyle\underline{L}\phi-\frac{\psi}{|u|}\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}(|\psi|+1).

To eliminate ψ𝜓\psi we should integrate this estimate over u𝑢u. For any small enough κ>0𝜅0\kappa>0,

(|u|κu0u|u|3κ|L¯ϕψ|u||2du)12κδ𝒲12𝒜.subscriptless-than-or-similar-to𝜅superscriptsuperscript𝑢𝜅superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢3𝜅superscript¯𝐿italic-ϕ𝜓superscript𝑢2differential-dsuperscript𝑢12𝛿superscript𝒲12𝒜\displaystyle\left(|u|^{\kappa}\int_{u_{0}}^{u}|u^{\prime}|^{3-\kappa}\left|\underline{L}\phi-\frac{\psi}{|u^{\prime}|}\right|^{2}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\lesssim_{\kappa}\delta\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (1.23)

This is the desired estimate777A slightly better bound for L¯ϕψ|u|¯𝐿italic-ϕ𝜓𝑢\underline{L}\phi-\frac{\psi}{|u|}, with 𝒲12superscript𝒲12\mathscr{W}^{\frac{1}{2}} dropped, is obtained in the actual proof. for L¯ϕψ|u|¯𝐿italic-ϕ𝜓𝑢\underline{L}\phi-\frac{\psi}{|u|}. It is crucial that κ=0𝜅0\kappa=0 leads to a logarithmic loss, which we will discuss later.

1.6.2. Reductive structure and the smallness condition

The presense of the borderline terms in the equations (1.20) requires that the estimates for ΘΘ\Theta should be obtained first without knowing any informations of Θ/\Theta\mkern-12.0mu/ and Θ¯¯Θ\underline{\Theta}. For example, the equation for Ωχ^Ω^𝜒\Omega\widehat{\chi} reads

D¯(Ωχ^)12Ωtrχ¯Ωχ^=Ω2(/ Θ/ +Θ/ Θ/ )+ΘΘ¯¯𝐷Ω^𝜒12Ωtr¯𝜒Ω^𝜒superscriptΩ2/ Θ/ Θ/ Θ/ Θ¯Θ\displaystyle\underline{D}(\Omega\widehat{\chi})-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\Omega\widehat{\chi}=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\mbox{$\Theta\mkern-12.0mu/$ }+\mbox{$\Theta\mkern-12.0mu/$ }\cdot\mbox{$\Theta\mkern-12.0mu/$ })+\Theta\cdot\underline{\Theta} (1.24)

If we assume that the estimates (1.19), (1.21) and (1.22) hold with the right hand side multiplied by a large constant C14superscript𝐶14C^{\frac{1}{4}}, then the above equation implies that

|u||Ωχ^||u0||Ωχ^||Cu0+Ω02(u0)(C14δ|u|1𝒲12𝒜+C12δ2|u|22𝒲𝒜2)+C12δ|u|12𝒜2.less-than-or-similar-to𝑢Ω^𝜒evaluated-atsubscript𝑢0Ω^𝜒subscript𝐶subscript𝑢0superscriptsubscriptΩ02subscript𝑢0superscript𝐶14𝛿superscript𝑢1superscript𝒲12𝒜superscript𝐶12superscript𝛿2superscript𝑢2superscript2𝒲superscript𝒜2superscript𝐶12𝛿superscript𝑢1superscript2superscript𝒜2\displaystyle|u||\Omega\widehat{\chi}|\lesssim|u_{0}||\Omega\widehat{\chi}|\Big{|}_{C_{u_{0}}}+\Omega_{0}^{2}(u_{0})(C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+C^{\frac{1}{2}}\delta^{2}|u|^{-2}\mathscr{F}^{2}\mathscr{W}\mathcal{A}^{2})+C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^{2}\mathcal{A}^{2}.

We then need the smallness condition (1.17) to obtain the desired estimate (1.19) for |u||u1|𝑢subscript𝑢1|u|\geq|u_{1}|. Indeed, by plugging in (1.17), the contribution from the right hand side becomes

C1𝒜superscript𝐶1𝒜\displaystyle C^{-1}\mathscr{F}\mathcal{A}

which is C1superscript𝐶1C^{-1} smaller than its expected estimate (1.19). We therefore find that the right hand side of the equation (1.24) contains no borderline terms, i.e., can be absorbed under the smallness condition (1.17). Having the estimates for ΘΘ\Theta, we are able to estimate Θ/\Theta\mkern-12.0mu/ , Θ¯¯Θ\underline{\Theta} and L¯ϕψ|u|¯𝐿italic-ϕ𝜓𝑢\underline{L}\phi-\frac{\psi}{|u|} using equations (1.20). Note that we also have many terms on the right hand sides of (1.20) which are not borderline and can be handled as above. Finally we will find that the estimates (1.19), (1.21) and (1.22) hold without C𝐶C and hence by a bootstrap argument, they really hold for u¯[0,δ],u[u0,u1]formulae-sequence¯𝑢0𝛿𝑢subscript𝑢0subscript𝑢1\underline{u}\in[0,\delta],u\in[u_{0},u_{1}].

The above argument shows that the Einstein equations have certain reductive structure. It means that the estimates for all related quantities can be divided into several steps. Although the error terms of the equations are nonlinear and highly coupled, either they can be absorbed by a suitable smallness condition, in which case they are non-borderline, or they only involve factors which are already estimated in previous steps.

1.6.3. Energy estimates for the curvature and the renormalization of the null Bianchi equaitons

The estimates for the connection coefficients should be coupled with the energy estimates for the curvature components. This is because the null structure equations which we use to estimate the connection coefficients contain curvature terms viewed as sources. The energy estimates for the curvature are based on the contracted Bianchi equations. Instead of introducing the Bel-Robinson tensor as in [9, 10], we will simply do integration by parts based on the so-called null Bianchi equations which are equations about the null components of the Weyl tensor and obtained by decompose the contracted Bianchi equations along null directions. The advantage of this procedure over introducing the Bel-Robinson tensor is that, in the case when some of the null components of the Weyl curvature cannot be controlled due to the nature of the problem, we can still do renormalization to the null Bianchi equations such that we are able to estimate some quantities instead of the Weyl curvature. This technique was first introduced by Luk-Rodnianski in [21] and subsequently developed or applied in [2, 17, 22, 20].

The problem we study shares many common features with the work [2]. Instead of estimating the full set of the null components of the Weyl curvature, they estimated β,K|u|2,σˇ=σ12χ^χ¯^𝛽𝐾superscript𝑢2ˇ𝜎𝜎12^𝜒^¯𝜒\beta,K-|u|^{-2},\check{\sigma}=\sigma-\frac{1}{2}\widehat{\chi}\wedge\widehat{\underline{\chi}} and β¯¯𝛽\underline{\beta} where K𝐾K is the Gauss curvature of Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u}. Recalling we couple the Einstein equations with the scalar field, the null Codazzi equation for div/ (Ωχ^)div/ Ω^𝜒\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi}) reads

div/ (Ωχ^)=(ΩβLϕ/ ϕ)div/ Ω^𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})=\cdots-(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)

Then ΩβΩ𝛽\Omega\beta is expected to behave like |u|2𝒜superscript𝑢2𝒜|u|^{-2}\mathscr{F}\mathcal{A}. In the procedure of the energy estimates, the estimates of β𝛽\beta (and its derivatives) along Cusubscript𝐶𝑢C_{u} should be done together with the estimates of K|u|2𝐾superscript𝑢2K-|u|^{-2} and σˇˇ𝜎\check{\sigma} along C¯u¯subscript¯𝐶¯𝑢\underline{C}_{\underline{u}}. The corresponding group of the renormalized null Bianchi equations are

D¯(Ωβ)+12Ωtrχ¯Ωβ=¯𝐷Ω𝛽12Ωtr¯𝜒Ω𝛽absent\displaystyle\underline{D}(\Omega\beta)+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\Omega\beta= Ω2(/ K/ σˇ)+,superscriptΩ2/ 𝐾superscript/ ˇ𝜎\displaystyle-\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }K-{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma})+\cdots,
D(K1|u|2)=𝐷𝐾1superscript𝑢2absent\displaystyle D\left(K-\frac{1}{|u|^{-2}}\right)= div/ (Ωβ)+,div/ Ω𝛽\displaystyle-\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\beta)+\cdots,
Dσˇ=𝐷ˇ𝜎absent\displaystyle D\check{\sigma}= curl/ (Ωβ)+.curl/ Ω𝛽\displaystyle-\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\beta)+\cdots.

The first equation should be written in the form D¯(Ωβ)¯𝐷Ω𝛽\underline{D}(\Omega\beta) but not D¯β¯𝐷𝛽\underline{D}\beta because the latter one will involve ω¯¯𝜔\underline{\omega}, the same as the equation for D¯(Ωχ^)¯𝐷Ω^𝜒\underline{D}(\Omega\widehat{\chi}) which is mentioned in Remark 1.8. We multiply the first by |u|2Ωβsuperscript𝑢2Ω𝛽|u|^{2}\Omega\beta, the second by |u|2Ω2(K|u|2)superscript𝑢2superscriptΩ2𝐾superscript𝑢2|u|^{2}\Omega^{2}(K-|u|^{-2}) and third by |u|2Ω2σˇsuperscript𝑢2superscriptΩ2ˇ𝜎|u|^{2}\Omega^{2}\check{\sigma}, and sum them up. Integrating the resulting equation in the spacetime, we can then expect

0δSu¯,u|u|2|Ωβ|2dμg/du¯+u0uΩ02(u)|u|2Su¯,u(|K1|u|2|2+|σˇ|2)dμg/duδ2𝒜2.\displaystyle\int_{0}^{\delta}\int_{S_{\underline{u},u}}|u|^{2}|\Omega\beta|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\mathrm{d}\underline{u}+\int_{u_{0}}^{u}\Omega_{0}^{2}(u^{\prime})|u^{\prime}|^{2}\int_{S_{\underline{u},u^{\prime}}}\left(\left|K-\frac{1}{|u^{\prime}|^{2}}\right|^{2}+|\check{\sigma}|^{2}\right)\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\mathrm{d}u^{\prime}\lesssim\delta\mathscr{F}^{2}\mathcal{A}^{2}.

Here Ω0subscriptΩ0\Omega_{0} appears again as a weight function. The other group of the renormalized equations are the following:

D¯(K1|u|2)+32Ωtrχ¯(K1|u|2)=¯𝐷𝐾1superscript𝑢232Ωtr¯𝜒𝐾1superscript𝑢2absent\displaystyle\underline{D}\left(K-\frac{1}{|u|^{2}}\right)+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}\left(K-\frac{1}{|u|^{2}}\right)= div/ (Ωβ¯)+,div/ Ω¯𝛽\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\underline{\beta})+\cdots,
D¯σˇ+32Ωtrχ¯σˇ=¯𝐷ˇ𝜎32Ωtr¯𝜒ˇ𝜎absent\displaystyle\underline{D}\check{\sigma}+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}\check{\sigma}= curl/ (Ωβ¯)+,curl/ Ω¯𝛽\displaystyle-\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\underline{\beta})+\cdots,
D(Ωβ¯)=𝐷Ω¯𝛽absent\displaystyle D(\Omega\underline{\beta})= Ω2(/ K+/ σˇ)+L¯ϕ/ Lϕ+.superscriptΩ2/ 𝐾superscript/ ˇ𝜎¯𝐿italic-ϕ/ 𝐿italic-ϕ\displaystyle\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }K+{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma})+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi+\cdots.

The rule is again the D¯¯𝐷\underline{D} equations do not involve ω¯¯𝜔\underline{\omega}. According to the coefficient 3232\frac{3}{2} of the second terms of the first888It is one of the most important observations in [2] that this first equation can be written in the form that the coefficient of the linear part is 3232\frac{3}{2} and the terms denoted by \cdots are under control. The coefficient of the linear part of the original equation is 111. and second equations, we multiply the first equation by |u|4(K|u|2)superscript𝑢4𝐾superscript𝑢2|u|^{4}(K-|u|^{-2}), the second by |u|4σˇsuperscript𝑢4ˇ𝜎|u|^{4}\check{\sigma} and the third by |u|4Ω2(Ωβ¯)superscript𝑢4superscriptΩ2Ω¯𝛽|u|^{4}\Omega^{-2}(\Omega\underline{\beta}), sum them up and then integrate the resulting equation. Note that we have a large factor Ω2superscriptΩ2\Omega^{-2} when estimating the error terms coming from the third equation, we cannot expect that K|u|2𝐾superscript𝑢2K-|u|^{-2} and σˇˇ𝜎\check{\sigma} can be estimated without Ω0subscriptΩ0\Omega_{0} weighted. Instead, we can only hope to estimate the following

Ω02(u)0δSu¯,u|u|4(|K1|u|2|2+|σˇ|2)dμg/du¯\displaystyle\Omega_{0}^{2}(u)\int_{0}^{\delta}\int_{S_{\underline{u},u}}|u|^{4}\left(\left|K-\frac{1}{|u^{\prime}|^{2}}\right|^{2}+|\check{\sigma}|^{2}\right)\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\mathrm{d}\underline{u}

We will see in the proof that it is bounded by δ33𝒜3superscript𝛿3superscript3superscript𝒜3\delta^{3}\mathscr{F}^{3}\mathcal{A}^{3}. On the other hand, the term L¯ϕ/ Lϕ¯𝐿italic-ϕ/ 𝐿italic-ϕ\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi in the third equation is a deadly term. This is because only the angular derivatives of L¯ϕ¯𝐿italic-ϕ\underline{L}\phi have good enough behavior but L¯ϕ¯𝐿italic-ϕ\underline{L}\phi itself behaves only like its initial value ψ|u|𝜓𝑢\frac{\psi}{|u|} on C¯0subscript¯𝐶0\underline{C}_{0}. Therefore, we should write

L¯ϕ/ Lϕ=D(L¯ϕ/ ϕ)DL¯ϕ/ ϕ.¯𝐿italic-ϕ/ 𝐿italic-ϕ𝐷¯𝐿italic-ϕ/ italic-ϕ𝐷¯𝐿italic-ϕ/ italic-ϕ\displaystyle\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi=D(\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-D\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi.

Then the first term can be absorbed by considering the null Bianchi equation for D(Ωβ¯+L¯ϕ/ ϕ)𝐷Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕD(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) instead of D(Ωβ¯)𝐷Ω¯𝛽D(\Omega\underline{\beta}) and the second term can be controlled using the wave equation. As a consequence, only Ωβ¯+L¯ϕ/ ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi but not Ωβ¯Ω¯𝛽\Omega\underline{\beta} can be controlled in the norm

Ω02(u)u0uΩ02(u)|u|4Su¯,u|Ωβ¯+L¯ϕ/ ϕ|2dμg/du\displaystyle\Omega_{0}^{2}(u)\int_{u_{0}}^{u}\Omega_{0}^{-2}(u^{\prime})|u^{\prime}|^{4}\int_{S_{\underline{u},u^{\prime}}}|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\mathrm{d}u^{\prime}

which is bounded again by δ33𝒜3superscript𝛿3superscript3superscript𝒜3\delta^{3}\mathscr{F}^{3}\mathcal{A}^{3}. We should remark here that this estimate will cause a minor difficulty. Consider the equation

D¯η¯=+(Ωβ¯+L¯ϕ/ ϕ)2L¯ϕ/ ϕ¯𝐷¯𝜂Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ2¯𝐿italic-ϕ/ italic-ϕ\displaystyle\underline{D}\underline{\eta}=\cdots+(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-2\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi

where the last term contributes to the desired estimate for η¯¯𝜂\underline{\eta} and the terms denoted by \cdots are also under control. Integrating the above equation we will have

|u||η¯||u0||η¯||Cu0+u0u|u||Ωβ¯+L¯ϕ/ ϕ|du+δ|u|1𝒲12𝒜.less-than-or-similar-to𝑢¯𝜂evaluated-atsubscript𝑢0¯𝜂subscript𝐶subscript𝑢0superscriptsubscriptsubscript𝑢0𝑢superscript𝑢Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕdifferential-dsuperscript𝑢𝛿superscript𝑢1superscript𝒲12𝒜\displaystyle|u||\underline{\eta}|\lesssim|u_{0}||\underline{\eta}|\big{|}_{C_{u_{0}}}+\int_{u_{0}}^{u}|u^{\prime}||\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi|\mathrm{d}u^{\prime}+\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.

The second term on the right hand side should be estimated as, by Hölder inequality,

u0u|u||Ωβ¯+L¯ϕ/ ϕ|du(u0uΩ02(u)|u|4du)12(u0uΩ02(u)|u|6|Ωβ¯+L¯ϕ/ ϕ|2du)12.less-than-or-similar-tosuperscriptsubscriptsubscript𝑢0𝑢superscript𝑢Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕdifferential-dsuperscript𝑢superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02superscript𝑢superscriptsuperscript𝑢4differential-dsuperscript𝑢12superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02superscript𝑢superscriptsuperscript𝑢6superscriptΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ2differential-dsuperscript𝑢12\displaystyle\int_{u_{0}}^{u}|u^{\prime}||\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi|\mathrm{d}u^{\prime}\lesssim\left(\int_{u_{0}}^{u}\Omega_{0}^{2}(u^{\prime})|u^{\prime}|^{-4}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\left(\int_{u_{0}}^{u}\Omega_{0}^{-2}(u^{\prime})|u^{\prime}|^{6}|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}.

Now recalling Ω0subscriptΩ0\Omega_{0} is monotonically decreasing, the first factor on the right is bounded by Ω0(u0)|u|32subscriptΩ0subscript𝑢0superscript𝑢32\Omega_{0}(u_{0})|u|^{-\frac{3}{2}}, and the second factor is bounded by Ω01(u)δ3232𝒜32superscriptsubscriptΩ01𝑢superscript𝛿32superscript32superscript𝒜32\Omega_{0}^{-1}(u)\delta^{\frac{3}{2}}\mathcal{F}^{\frac{3}{2}}\mathcal{A}^{\frac{3}{2}} mentioned above. Then, the contribution of the term Ωβ¯+L¯ϕ/ ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi is

Ω0(u0)Ω01(u)(δ|u|1𝒜)32.subscriptΩ0subscript𝑢0superscriptsubscriptΩ01𝑢superscript𝛿superscript𝑢1𝒜32\displaystyle\Omega_{0}(u_{0})\Omega_{0}^{-1}(u)(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{3}{2}}.

We can see that only the condition (1.17) is not enough to make sure that η¯¯𝜂\underline{\eta} obeys the expected estimate. Therefore the auxiliary condition (1.18) is needed to ensure that the a priori estimates can be done.

1.6.4. Energy estimates for the scalar field and a logarithmic loss on the top order derivatives

Recalling we are coupling the Einstein equations with the massless scalar field ϕitalic-ϕ\phi which satisfying the wave equations, through which we can obtain estimates for the scalar field. We will not introduce the energy momentum tensor but simply do integration by parts using the wave equations written in the double null coordinate system:

D¯Lϕ+12Ωtrχ¯Lϕ¯𝐷𝐿italic-ϕ12Ωtr¯𝜒𝐿italic-ϕ\displaystyle\underline{D}L\phi+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi =Ω2Δ/ ϕ+2Ω2(η,/ ϕ)12ΩtrχL¯ϕ,absentsuperscriptΩ2Δ/ italic-ϕ2superscriptΩ2𝜂/ italic-ϕ12Ωtr𝜒¯𝐿italic-ϕ\displaystyle=\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}(\eta,\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi,
D/ ϕ𝐷/ italic-ϕ\displaystyle D\mbox{$\nabla\mkern-13.0mu/$ }\phi =/ Lϕ,absent/ 𝐿italic-ϕ\displaystyle=\mbox{$\nabla\mkern-13.0mu/$ }L\phi,
DL¯ϕ+12ΩtrχL¯ϕ𝐷¯𝐿italic-ϕ12Ωtr𝜒¯𝐿italic-ϕ\displaystyle D\underline{L}\phi+\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi =Ω2Δ/ ϕ+2Ω2(η¯,/ ϕ)12Ωtrχ¯Lϕ,absentsuperscriptΩ2Δ/ italic-ϕ2superscriptΩ2¯𝜂/ italic-ϕ12Ωtr¯𝜒𝐿italic-ϕ\displaystyle=\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}(\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi,
D¯/ ϕ¯𝐷/ italic-ϕ\displaystyle\underline{D}\mbox{$\nabla\mkern-13.0mu/$ }\phi =/ L¯ϕ.absent/ ¯𝐿italic-ϕ\displaystyle=\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi.

There are four equations here but the first and the third are exactly the same equation, which is the wave equation. The second and the fourth equations are simply geometric identities. Written in this form, the first two equations can be considered as a group of equations for Lϕ𝐿italic-ϕL\phi, / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi and the last two can be considered as a group of equations for / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi, L¯ϕ¯𝐿italic-ϕ\underline{L}\phi. They have the same structure as the null Bianchi equations and then the energy estimates can be done in the same way.

Now we want to discuss what will happen for the estimates for the top order derivatives of the geometric quantities. It is not surprising that the top order estimates are worse than the lower order estimates. We may think this is due to the nonlinear nature of the equations, because on a technical level, we use the transport equations to estimate the lower order derivatives, and use however the Hodge-transport coupled system to estimate the top order derivatives. And sometimes different approaches give arise to different estimates. For example, similar to [2], the estimate along C¯u¯subscript¯𝐶¯𝑢\underline{C}_{\underline{u}} for the top order derivative of η¯¯𝜂\underline{\eta} is worse than η𝜂\eta. Besides this, we will have a logarithmic loss due to the presence of the scalar field. Let us look at the estimate for Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi}. Its lower order derivatives can be estimated easily by the equation

D(Ωtrχ¯)=Ω2/ η+.𝐷Ωtr¯𝜒superscriptΩ2/ 𝜂\displaystyle D(\Omega\mathrm{tr}\underline{\chi})=\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\eta+\cdots.

However, integrating this equation will cause a loss of derivative and therefore we cannot use this equation to estimate the top order derivative of Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi}, which should be estimated using the equation for D¯(Ωtrχ¯)¯𝐷Ωtr¯𝜒\underline{D}(\Omega\mathrm{tr}\underline{\chi}) coupled with the null Codazzi equation for div/ (Ωχ¯^)div/ Ω^¯𝜒\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}}). Because we only hope to describe here the main ideas, for convenience we suppose that we only estimate / (Ωtrχ¯)/ Ωtr¯𝜒\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}) itself using these equations, although we know that the lower order estimates can be obtained much more easily. Now the equation for D¯(Ωtrχ¯)¯𝐷Ωtr¯𝜒\underline{D}(\Omega\mathrm{tr}\underline{\chi}) reads

D¯(Ωtrχ¯)=12(Ωtrχ¯)2+2ω¯Ωtrχ¯|Ωχ¯^|22(L¯ϕ)2¯𝐷Ωtr¯𝜒12superscriptΩtr¯𝜒22¯𝜔Ωtr¯𝜒superscriptΩ^¯𝜒22superscript¯𝐿italic-ϕ2\underline{D}(\Omega\mathrm{tr}\underline{\chi})=-\frac{1}{2}(\Omega\mathrm{tr}\underline{\chi})^{2}+2\underline{\omega}\Omega\mathrm{tr}\underline{\chi}-|\Omega\widehat{\underline{\chi}}|^{2}-2(\underline{L}\phi)^{2}

and applying an angular derivative /\nabla\mkern-13.0mu/ , we have

D¯(Ω2/ (Ωtrχ¯))+Ωtrχ¯(Ω2/ (Ωtrχ¯))=2Ω2/ ω¯(Ωtrχ¯)2Ω2(Ωχ¯^)/ (Ωχ¯^)4Ω2L¯ϕ/ L¯ϕ.¯𝐷superscriptΩ2/ Ωtr¯𝜒Ωtr¯𝜒superscriptΩ2/ Ωtr¯𝜒2superscriptΩ2/ ¯𝜔Ωtr¯𝜒2superscriptΩ2Ω^¯𝜒/ Ω^¯𝜒4superscriptΩ2¯𝐿italic-ϕ/ ¯𝐿italic-ϕ\begin{split}&\underline{D}(\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}))+\Omega\mathrm{tr}\underline{\chi}(\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}))\\ &=2\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}(\Omega\mathrm{tr}\underline{\chi})-2\Omega^{-2}(\Omega\widehat{\underline{\chi}})\cdot\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}})-4\Omega^{-2}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi.\\ \end{split} (1.25)

Note that the equation is written in this form in order that it does not involve ω¯¯𝜔\underline{\omega} itself (but involves the angular derivatives of ω¯¯𝜔\underline{\omega}). The coefficient 111 of the second term on the left hand side suggests that the right hand side should be estimated in

u0u|u|3||du.\displaystyle\int_{u_{0}}^{u}|u^{\prime}|^{3}|\cdot|\mathrm{d}u^{\prime}.

From the equation999The top order estimate for ω¯¯𝜔\underline{\omega} should be derived using the transport-Hodge coupled system instead of this transport equation, but the term causing trouble is essentially the same.

D/ ω¯𝐷/ ¯𝜔\displaystyle D\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega} =L¯ϕ/ Lϕ+,absent¯𝐿italic-ϕ/ 𝐿italic-ϕ\displaystyle=-\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi+\cdots,

the first term on the right hand side of (1.25) is estimated by

u0u|u|3|Ω2/ ω¯(Ωtrχ¯)|dusuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢3superscriptΩ2/ ¯𝜔Ωtr¯𝜒differential-dsuperscript𝑢\displaystyle\int_{u_{0}}^{u}|u^{\prime}|^{3}|\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}(\Omega\mathrm{tr}\underline{\chi})|\mathrm{d}u^{\prime}
less-than-or-similar-to\displaystyle\lesssim Ω02(u)δ(u0u|ψ|2|u|du)12sup0u¯δ(u0u|u|3/ Lϕ𝕃(u¯,u)2du)12+.superscriptsubscriptΩ02𝑢𝛿superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢12subscriptsupremum0¯𝑢𝛿superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢3superscriptsubscriptnorm/ 𝐿italic-ϕsuperscript𝕃¯𝑢superscript𝑢2differential-dsuperscript𝑢12\displaystyle\Omega_{0}^{-2}(u)\delta\left(\int_{u_{0}}^{u}\frac{|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\sup_{0\leq\underline{u}\leq\delta}\left(\int_{u_{0}}^{u}|u^{\prime}|^{3}\|\mbox{$\nabla\mkern-13.0mu/$ }L\phi\|_{\mathbb{L}^{\infty}(\underline{u},u^{\prime})}^{2}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}+\cdots.

From the equation

D/ L¯ϕ𝐷/ ¯𝐿italic-ϕ\displaystyle D\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi =12Ωtrχ¯/ Lϕ+.absent12Ωtr¯𝜒/ 𝐿italic-ϕ\displaystyle=-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\mbox{$\nabla\mkern-13.0mu/$ }L\phi+\cdots.

The last term of (1.25) can be estimated similarly by

u0u|u|3|Ω2L¯ϕ/ L¯ϕ|dusuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢3superscriptΩ2¯𝐿italic-ϕ/ ¯𝐿italic-ϕdifferential-dsuperscript𝑢\displaystyle\int_{u_{0}}^{u}|u^{\prime}|^{3}|\Omega^{-2}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi|\mathrm{d}u^{\prime}
less-than-or-similar-to\displaystyle\lesssim Ω02(u)δ(u0u|ψ|2|u|du)12sup0u¯δ(u0u|u|3/ Lϕ𝕃(u¯,u)2du)12+.superscriptsubscriptΩ02𝑢𝛿superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢12subscriptsupremum0¯𝑢𝛿superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢3superscriptsubscriptnorm/ 𝐿italic-ϕsuperscript𝕃¯𝑢superscript𝑢2differential-dsuperscript𝑢12\displaystyle\Omega_{0}^{-2}(u)\delta\left(\int_{u_{0}}^{u}\frac{|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\sup_{0\leq\underline{u}\leq\delta}\left(\int_{u_{0}}^{u}|u^{\prime}|^{3}\|\mbox{$\nabla\mkern-13.0mu/$ }L\phi\|_{\mathbb{L}^{\infty}(\underline{u},u^{\prime})}^{2}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}+\cdots.

We can see both estimates cannot be controlled without a logarithmic loss101010Observe however that the term Ωtrχ¯L¯ϕ/ LϕΩtr¯𝜒¯𝐿italic-ϕ/ 𝐿italic-ϕ\Omega\mathrm{tr}\underline{\chi}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi, which we are dealing with, disappears in the equation for D(/ ω¯(Ωtrχ¯)2L¯ϕ/ L¯ϕ)𝐷/ ¯𝜔Ωtr¯𝜒2¯𝐿italic-ϕ/ ¯𝐿italic-ϕD(\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}(\Omega\mathrm{tr}\underline{\chi})-2\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi). We could have killed the logarithmic loss by using this cancellation, which unfortunately requires a loss of derivative and only works for the lower order estimates. because / Lϕ/ 𝐿italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }L\phi can only be bounded by |u|2𝒜superscript𝑢2𝒜|u|^{-2}\mathscr{F}\mathcal{A}. The worst thing that could happen is that this loss would accumulate and the bootstrap argument cannot be closed. Fortunately, / Lϕ/ 𝐿italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }L\phi satisfies a better estimate:

|u||/ Lϕ||u0||/ Lϕ||Cu0+δ|u|12𝒲12𝒜2.less-than-or-similar-to𝑢/ 𝐿italic-ϕevaluated-atsubscript𝑢0/ 𝐿italic-ϕsubscript𝐶subscript𝑢0𝛿superscript𝑢1superscript2superscript𝒲12superscript𝒜2\displaystyle|u||\mbox{$\nabla\mkern-13.0mu/$ }L\phi|\lesssim|u_{0}||\mbox{$\nabla\mkern-13.0mu/$ }L\phi|\big{|}_{C_{u_{0}}}+\delta|u|^{-1}\mathscr{F}^{2}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}.

The second term on the right hand side will not suffer the logarithmic loss by first integrating along C¯u¯subscript¯𝐶¯𝑢\underline{C}_{\underline{u}} and then plugging in the smallness condition (1.17). The loss now only depends on the initial value of / Lϕ/ 𝐿italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }L\phi. So we introduce the quantity \mathscr{E} defined in (1.14) and we will find that the top order estimates are expected to suffer a loss \mathscr{E}. Moreover, the next to top order derivatives of η,η¯𝜂¯𝜂\eta,\underline{\eta}, / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi and all curvature components also suffer such a loss. The reason is that, for example, the estimate for the next to top order estimates of η𝜂\eta, η¯¯𝜂\underline{\eta} and / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi are derived in terms of the top order derivatives of β𝛽\beta, β¯+L¯ϕ/ ϕ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi and Lϕ𝐿italic-ϕL\phi respectively. Recalling in the statement of Theorem 1.12 that we will estimate up to the \engordnumber4 order derivatives of the curvature and up to the \engordnumber5 order derivatives of the connection coefficients and the derivatives of the scalar field function ϕitalic-ϕ\phi. Then it is only the \engordnumber3 and lower order estimates do not suffer this loss. We remark that in proving Theorem 1.6, the formation of trapped surface, Lsuperscript𝐿L^{\infty} estimate without logarithmic loss for the / η/ 𝜂\mbox{$\nabla\mkern-13.0mu/$ }\eta is needed, therefore the estimates for up to the \engordnumber5 order derivatives of the connection coefficients are needed.

Although only the top and next to top estimates have the loss, we find that it is convenient to only derive estimates with loss for all derivatives of η,η¯,/ ϕ𝜂¯𝜂/ italic-ϕ\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi in the proof of existence. The expected estimate obeyed by Θ/ {η,η¯,/ ϕ}Θ/ 𝜂¯𝜂/ italic-ϕ\mbox{$\Theta\mkern-12.0mu/$ }\in\{\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi\} then becomes

Θ/ δ|u|1𝒲12𝒜.less-than-or-similar-toΘ/ 𝛿superscript𝑢1superscript𝒲12𝒜\displaystyle\mbox{$\Theta\mkern-12.0mu/$ }\lesssim\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.

Because of this loss, an addition smallness condition should be introduced. Consider for example the equation for D(Ωχ¯^)𝐷Ω^¯𝜒D(\Omega\widehat{\underline{\chi}}) in the form

D(Ωχ¯^)=Ω2(/ η¯+η¯η¯)+𝐷Ω^¯𝜒superscriptΩ2/ ¯𝜂¯𝜂¯𝜂\displaystyle D(\Omega\widehat{\underline{\chi}})=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\underline{\eta}+\underline{\eta}\cdot\underline{\eta})+\cdots

The top order derivative for η¯¯𝜂\underline{\eta} can only be bounded in a scale invariant version of the norm L2(Cu)superscript𝐿2subscript𝐶𝑢L^{2}(C_{u}) by δ12|u|32𝒜superscript𝛿12superscript𝑢32𝒜\delta^{\frac{1}{2}}|u|^{-\frac{3}{2}}\mathscr{F}\mathscr{E}\mathcal{A}, with a loss (δ|u|1)12superscript𝛿superscript𝑢112(\delta|u|^{-1})^{\frac{1}{2}} as compared to its lower order estimates. Therefore, by integrating the above equation, we have,

|Ωχ¯^|Ω02(u)(δ32|u|52𝒲12𝒜+δ3|u|422𝒲𝒜2)+.less-than-or-similar-toΩ^¯𝜒superscriptsubscriptΩ02𝑢superscript𝛿32superscript𝑢52superscript𝒲12𝒜superscript𝛿3superscript𝑢4superscript2superscript2𝒲superscript𝒜2\displaystyle|\Omega\widehat{\underline{\chi}}|\lesssim\Omega_{0}^{2}(u)(\delta^{\frac{3}{2}}|u|^{-\frac{5}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+\delta^{3}|u|^{-4}\mathscr{F}^{2}\mathscr{E}^{2}\mathscr{W}\mathcal{A}^{2})+\cdots.

The desired estimate can then be obtained by introducing another smallness condition (1.16).

Remark 1.13.

Strictly speaking, the condition (1.16) has nothing to do with the smallness because we do not need to require that the quantity on the left hand side is very small. But we still name it the smallness contidion because it is used very frequently as the real smallness condition (1.17). An unsatisfactory consequence is that when considering the next to top order estimate for Ωχ¯^Ω^¯𝜒\Omega\widehat{\underline{\chi}}, the first term Ω2/ η¯superscriptΩ2/ ¯𝜂\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\underline{\eta} on the right hand side of the equation for D(Ωχ¯^)𝐷Ω^¯𝜒D(\Omega\widehat{\underline{\chi}}) becomes borderline. Nevertheless we can also close the estimates because the top order estimate for η¯¯𝜂\underline{\eta} do not contain borderline terms involving Ωχ¯^Ω^¯𝜒\Omega\widehat{\underline{\chi}}.

We then close the discussion on the proof of the existence theorem.

1.7. Outline of the paper

The remainder of this paper is organized as follows. In Section 2, we introduce the double null coordinate system, and the notations and equations adapted to this coordinate system we will use in the proof. In Section 3, we will state the existence theorem, whose proof is then given in Section 4. The formation of trapped surfaces theorems are proven in Section 5 and the instability theorems are proven in Section LABEL:Secinstability. In the last section, Section 6, we will state and prove the precise version of the Theorem 1.4, the main result of this paper.

1.8. Acknowledgement

Both authors are supported by NSFC 11501582, 11521101. The first author would like to thank Jonathan Luk for valuable discussions on this topic and suggestions on the first version of the manuscript. He would also like to thank Xinliang An for showing his interest in this work. Both authors would like to thank Xi-Ping Zhu for his supports, discussions and continuous encouragements.

2. Double null coordinate system and equations

2.1. Double null coordinate system

We first introduce the geometric setup. We use (M,g)𝑀𝑔(M,g) to denote the underlying space-time (which will be the solution) with the Lorentzian metric and use \nabla to denote the Levi-Civita connection of the metric g𝑔g. Let u𝑢u and u¯¯𝑢\underline{u} be two optical functions on M𝑀M, that is

g(u,u)=g(u¯,u¯)=0.𝑔𝑢𝑢𝑔¯𝑢¯𝑢0g(\nabla u,\nabla u)=g(\nabla\underline{u},\nabla\underline{u})=0.

The space-time M𝑀M is foliated by the level sets of u¯¯𝑢\underline{u} and u𝑢u respectively, and the functions u𝑢u and u¯¯𝑢\underline{u} are required to increase towards the future. We use Cusubscript𝐶𝑢C_{u} to denote the outgoing null hypersurfaces which are the level sets of u𝑢u and use C¯u¯subscript¯𝐶¯𝑢{\underline{C}}_{\underline{u}} to denote the incoming null hypersurfaces which are the level sets of u¯¯𝑢\underline{u}. We denote the intersection Su¯,u=C¯u¯Cusubscript𝑆¯𝑢𝑢subscript¯𝐶¯𝑢subscript𝐶𝑢S_{\underline{u},u}=\underline{C}_{\underline{u}}\cap C_{u}, which is a space-like two-sphere.

We define a positive function ΩΩ\Omega by the formula

Ω2=2g(u¯,u).superscriptΩ22𝑔¯𝑢𝑢\Omega^{-2}=-2g(\nabla\underline{u},\nabla u).

We then define the normalized null pair (L¯^,L^)^¯𝐿^𝐿(\widehat{\underline{L}},\widehat{L}) by

L¯^=2Ωu¯,L^=2Ωu,formulae-sequence^¯𝐿2Ω¯𝑢^𝐿2Ω𝑢\widehat{\underline{L}}=-2\Omega\nabla\underline{u},\ \widehat{L}=-2\Omega\nabla u,

which also be denoted by e3,e4subscript𝑒3subscript𝑒4e_{3},e_{4} respectively, and define one another null pair

L¯=ΩL¯^,L=ΩL^.formulae-sequence¯𝐿Ω^¯𝐿𝐿Ω^𝐿\underline{L}=\Omega\widehat{\underline{L}},\ L=\Omega\widehat{L}.

The flows generated by L¯¯𝐿\underline{L} and L𝐿L preserve the double null foliation. On a given two sphere Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u} we choose a local frame e1,e2subscript𝑒1subscript𝑒2{e_{1},e_{2}}. We call e1,e2,L¯^,L^subscript𝑒1subscript𝑒2^¯𝐿^𝐿{e_{1},e_{2},\widehat{\underline{L}},\widehat{L}} a null frame. As a convention, throughout the paper, we use capital Latin letters A,B,C,𝐴𝐵𝐶A,B,C,\cdots to denote an index from 111 to 222 and Greek letters α,β,𝛼𝛽\alpha,\beta,\cdots to denote an index from 111 to 444, e.g. eAsubscript𝑒𝐴e_{A} denotes either e1subscript𝑒1e_{1} or e2subscript𝑒2e_{2}.

We define ψ𝜓\psi to be a tangential tensorfield if ψ𝜓\psi is a priori a tensorfield defined on the space-time M𝑀M and all the possible contractions of ψ𝜓\psi with either L¯^^¯𝐿\widehat{\underline{L}} or L^^𝐿\widehat{L} are zeros. We use Dψ𝐷𝜓D\psi and D¯ψ¯𝐷𝜓\underline{D}\psi to denote the projection to Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u} of usual Lie derivatives Lψsubscript𝐿𝜓\mathcal{L}_{L}\psi and L¯ψsubscript¯𝐿𝜓\mathcal{L}_{\underline{L}}\psi. The space-time metric g𝑔g induces a Riemannian metric g/g\mkern-9.0mu/ on Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u} and ϵ/\epsilon\mkern-9.0mu/ is the volume form of g/g\mkern-9.0mu/ on Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u}. We use /\nabla\mkern-13.0mu/ and /\nabla\mkern-13.0mu/ to denote the exterior differential and covariant derivative (with respect to g/g\mkern-9.0mu/) on Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u}.

Using these two optical functions, we can introduce a local coordinate system (u¯,u,ϑA)¯𝑢𝑢superscriptitalic-ϑ𝐴(\underline{u},u,\vartheta^{A}) on M𝑀M. In such a coordinate system, the Lorentzian metric g𝑔g takes the following form

g=2Ω2(du¯du+dudu¯)+g/AB(dϑAbAdu¯)(dϑBbBdu¯),\displaystyle g=-2\Omega^{2}(\mathrm{d}\underline{u}\otimes\mathrm{d}u+\mathrm{d}u\otimes\mathrm{d}\underline{u})+\mbox{$g\mkern-9.0mu/$}_{AB}(\mathrm{d}\vartheta^{A}-b^{A}\mathrm{d}\underline{u})\otimes(\mathrm{d}\vartheta^{B}-b^{B}\mathrm{d}\underline{u}),

such that L¯¯𝐿\underline{L} and L𝐿L can be computed as L¯=u+bAϑA¯𝐿subscript𝑢superscript𝑏𝐴subscriptsuperscriptitalic-ϑ𝐴\underline{L}=\partial_{u}+b^{A}\partial_{\vartheta^{A}} and L=u¯𝐿subscript¯𝑢L=\partial_{\underline{u}}. We can also require that bAsuperscript𝑏𝐴b^{A} vanishes on some specific null cone.

We recall the null decomposition of the connection coefficients using the null frame (e1,e2,L¯^,L^)subscript𝑒1subscript𝑒2^¯𝐿^𝐿(e_{1},e_{2},\widehat{\underline{L}},\widehat{L}) as follows:

χABsubscript𝜒𝐴𝐵\displaystyle\chi_{AB} =g(AL^,eB),ηA=12g(L¯^eA,L^),ω=12Ωg(L^L¯^,L^),formulae-sequenceabsent𝑔subscript𝐴^𝐿subscript𝑒𝐵formulae-sequencesubscript𝜂𝐴12𝑔subscript^¯𝐿subscript𝑒𝐴^𝐿𝜔12Ω𝑔subscript^𝐿^¯𝐿^𝐿\displaystyle=g(\nabla_{A}\widehat{L},e_{B}),\quad\eta_{A}=-\frac{1}{2}g(\nabla_{\widehat{\underline{L}}}e_{A},\widehat{L}),\quad\omega=\frac{1}{2}\Omega g(\nabla_{\widehat{L}}\widehat{\underline{L}},\widehat{L}),
χ¯ABsubscript¯𝜒𝐴𝐵\displaystyle\underline{\chi}_{AB} =g(AL¯^,eB),η¯A=12g(L^eA,L¯^),ω¯=12Ωg(L¯^L^,L¯^).formulae-sequenceabsent𝑔subscript𝐴^¯𝐿subscript𝑒𝐵formulae-sequencesubscript¯𝜂𝐴12𝑔subscript^𝐿subscript𝑒𝐴^¯𝐿¯𝜔12Ω𝑔subscript^¯𝐿^𝐿^¯𝐿\displaystyle=g(\nabla_{A}\widehat{\underline{L}},e_{B}),\quad\underline{\eta}_{A}=-\frac{1}{2}g(\nabla_{\widehat{L}}e_{A},\widehat{\underline{L}}),\quad\underline{\omega}=\frac{1}{2}\Omega g(\nabla_{\widehat{\underline{L}}}\widehat{L},\widehat{\underline{L}}).

They are all tangential tensorfields. We also define the following normalized quantities:

χ=Ω1χ,χ¯=Ω1χ,ζ=12(ηη¯).formulae-sequencesuperscript𝜒superscriptΩ1𝜒formulae-sequencesuperscript¯𝜒superscriptΩ1𝜒𝜁12𝜂¯𝜂\chi^{\prime}=\Omega^{-1}\chi,\ \underline{\chi}^{\prime}=\Omega^{-1}\chi,\ \zeta=\frac{1}{2}(\eta-\underline{\eta}).

The trace of χ𝜒\chi and χ¯¯𝜒\underline{\chi} are denoted by

trχ=g/ABχAB,trχ¯=g/ABχ¯AB.\mathrm{tr}\chi=\mbox{$g\mkern-9.0mu/$}^{AB}\chi_{AB},\ \mathrm{tr}\underline{\chi}=\mbox{$g\mkern-9.0mu/$}^{AB}\underline{\chi}_{AB}.

By definition, we can check directly the following useful identities :

/ logΩ=12(η+η¯),DlogΩ=ω,D¯logΩ=ω¯.formulae-sequence/ Ω12𝜂¯𝜂formulae-sequence𝐷Ω𝜔¯𝐷Ω¯𝜔\mbox{$\nabla\mkern-13.0mu/$ }\log\Omega=\frac{1}{2}(\eta+\underline{\eta}),\ D\log\Omega=\omega,\ \underline{D}\log\Omega=\underline{\omega}.

We can also define the null components of the Weyl curvature tensor W:

αABsubscript𝛼𝐴𝐵\displaystyle\alpha_{AB} =𝐖(eA,L^,eB,L^),βA=12𝐖(eA,L^,L¯^,L^),ρ=14𝐖(L¯^,L^,L¯^,L^),formulae-sequenceabsent𝐖subscript𝑒𝐴^𝐿subscript𝑒𝐵^𝐿formulae-sequencesubscript𝛽𝐴12𝐖subscript𝑒𝐴^𝐿^¯𝐿^𝐿𝜌14𝐖^¯𝐿^𝐿^¯𝐿^𝐿\displaystyle=\mathbf{W}(e_{A},\widehat{L},e_{B},\widehat{L}),\quad\beta_{A}=\frac{1}{2}\mathbf{W}(e_{A},\widehat{L},\widehat{\underline{L}},\widehat{L}),\quad\rho=\frac{1}{4}\mathbf{W}(\widehat{\underline{L}},\widehat{L},\widehat{\underline{L}},\widehat{L}),
α¯ABsubscript¯𝛼𝐴𝐵\displaystyle\underline{\alpha}_{AB} =𝐖(eA,L¯^,eB,L¯^),β¯A=12𝐖(eA,L¯^,L¯^,L^),σ=14𝐖(L¯^,L^,eA,eB)ϵ/AB.\displaystyle=\mathbf{W}(e_{A},\widehat{\underline{L}},e_{B},\widehat{\underline{L}}),\quad\underline{\beta}_{A}=\frac{1}{2}\mathbf{W}(e_{A},\widehat{\underline{L}},\widehat{\underline{L}},\widehat{L}),\quad\sigma=\frac{1}{4}\mathbf{W}(\widehat{\underline{L}},\widehat{L},e_{A},e_{B})\mbox{$\epsilon\mkern-9.0mu/$}^{AB}.

2.2. Equations

Before we write down the equations we will use, we first define several kinds of contraction of the tangential tensorfields. For a symmetric tangential 2-tensorfield θ𝜃\theta, we use θ^^𝜃\widehat{\theta} and trθtr𝜃\mathrm{tr}\theta to denote the trace-free part and trace of θ𝜃\theta (with respect to g/g\mkern-9.0mu/). If θ𝜃\theta is trace-free, D^θ^𝐷𝜃\widehat{D}\theta and D¯^θ^¯𝐷𝜃\widehat{\underline{D}}\theta refer to the trace-free part of Dθ𝐷𝜃D\theta and D¯θ¯𝐷𝜃\underline{D}\theta. Let ξ𝜉\xi be a tangential 111-form. We define some products and operators for later use. For the products, we define

(θ1,θ2)=g/ACg/BD(θ1)AB(θ2)CD,(ξ1,ξ2)=g/AB(ξ1)A(ξ2)B.(\theta_{1},\theta_{2})=\mbox{$g\mkern-9.0mu/$}^{AC}\mbox{$g\mkern-9.0mu/$}^{BD}(\theta_{1})_{AB}(\theta_{2})_{CD},\ \ (\xi_{1},\xi_{2})=\mbox{$g\mkern-9.0mu/$}^{AB}(\xi_{1})_{A}(\xi_{2})_{B}.

This also leads to the following norms

|θ|2=(θ,θ),|ξ|2=(ξ,ξ).formulae-sequencesuperscript𝜃2𝜃𝜃superscript𝜉2𝜉𝜉|\theta|^{2}=(\theta,\theta),\ |\xi|^{2}=(\xi,\xi).

We then define the contractions

(θξ)A=θAξBBsubscript𝜃𝜉𝐴subscript𝜃𝐴superscriptsubscript𝜉𝐵𝐵\displaystyle(\theta\cdot\xi)_{A}=\theta_{A}{}^{B}\xi_{B} ,(θ1θ2)AB=(θ1)A(θ2)CBC,\displaystyle,\ (\theta_{1}\cdot\theta_{2})_{AB}=(\theta_{1})_{A}{}^{C}(\theta_{2})_{CB},
θ1θ2=ϵ/ACg/BD(θ1)AB(θ2)CD\displaystyle\theta_{1}\wedge\theta_{2}=\mbox{$\epsilon\mkern-9.0mu/$}^{AC}\mbox{$g\mkern-9.0mu/$}^{BD}(\theta_{1})_{AB}(\theta_{2})_{CD} ,ξ1^ξ2=ξ1ξ2+ξ2ξ1(ξ1,ξ2)g/.\displaystyle,\ \xi_{1}\widehat{\otimes}\xi_{2}=\xi_{1}\otimes\xi_{2}+\xi_{2}\otimes\xi_{1}-(\xi_{1},\xi_{2})\mbox{$g\mkern-9.0mu/$}.

The Hodge dual for ξ𝜉\xi is defined by ξA=ϵ/AξCC\prescript{*}{}{\xi}_{A}=\mbox{$\epsilon\mkern-9.0mu/$}_{A}{}^{C}\xi_{C}. For the operators, we define

div/ ξ=/ AξA,curl/ ξA=ϵ/AB/ AξB,(div/ θ)A=/ BθAB.\mbox{$\mathrm{div}\mkern-13.0mu/$ }\xi=\mbox{$\nabla\mkern-13.0mu/$ }^{A}\xi_{A},\ \mbox{$\mathrm{curl}\mkern-13.0mu/$ }\xi_{A}=\mbox{$\epsilon\mkern-9.0mu/$}^{AB}\mbox{$\nabla\mkern-13.0mu/$ }_{A}\xi_{B},\ (\mbox{$\mathrm{div}\mkern-13.0mu/$ }\theta)_{A}=\mbox{$\nabla\mkern-13.0mu/$ }^{B}\theta_{AB}.

We finally define a traceless operator

(/ ^ξ)AB=(/ ξ)AB+(/ ξ)BAdiv/ ξg/AB.(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\xi)_{AB}=(\mbox{$\nabla\mkern-13.0mu/$ }\xi)_{AB}+(\mbox{$\nabla\mkern-13.0mu/$ }\xi)_{BA}-\mbox{$\mathrm{div}\mkern-13.0mu/$ }\xi\,\mbox{$g\mkern-9.0mu/$}_{AB}.

The followings are the null structure equations (where K𝐾K is the Gauss curvature of Su¯,usubscript𝑆¯𝑢𝑢S_{\underline{u},u}):111111See Chapter 1 of [9] for the derivation of these equations in the vacuum case.

D¯(Ωtrχ¯)¯𝐷Ωtr¯𝜒\displaystyle\underline{D}(\Omega\mathrm{tr}\underline{\chi}) =12(Ωtrχ¯)2+2ω¯Ωtrχ¯|Ωχ¯^|22(L¯ϕ)2,absent12superscriptΩtr¯𝜒22¯𝜔Ωtr¯𝜒superscriptΩ^¯𝜒22superscript¯𝐿italic-ϕ2\displaystyle=-\frac{1}{2}(\Omega\mathrm{tr}\underline{\chi})^{2}+2\underline{\omega}\Omega\mathrm{tr}\underline{\chi}-|\Omega\widehat{\underline{\chi}}|^{2}-2(\underline{L}\phi)^{2},
Dtrχ𝐷trsuperscript𝜒\displaystyle D\mathrm{tr}\chi^{\prime} =12(Ωtrχ)2|χ^|22(L^ϕ)2,absent12superscriptΩtrsuperscript𝜒2superscript^𝜒22superscript^𝐿italic-ϕ2\displaystyle=-\frac{1}{2}(\Omega\mathrm{tr}\chi^{\prime})^{2}-|\widehat{\chi}|^{2}-2(\widehat{L}\phi)^{2},
D¯^(Ωχ^)^¯𝐷Ω^𝜒\displaystyle\widehat{\underline{D}}(\Omega\widehat{\chi}) =Ω2(/ ^η+η^η+12trχ¯χ^12trχχ¯^+/ ϕ^/ ϕ),absentsuperscriptΩ2/ ^tensor-product𝜂𝜂^tensor-product𝜂12tr¯𝜒^𝜒12tr𝜒^¯𝜒/ italic-ϕ^tensor-product/ italic-ϕ\displaystyle=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\eta+\eta\widehat{\otimes}\eta+\frac{1}{2}\mathrm{tr}\underline{\chi}\widehat{\chi}-\frac{1}{2}\mathrm{tr}\chi\widehat{\underline{\chi}}+\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi),
D^(Ωχ¯^)^𝐷Ω^¯𝜒\displaystyle\widehat{D}(\Omega\widehat{\underline{\chi}}) =Ω2(/ ^η¯+η¯^η¯+12trχχ¯^12trχ¯χ^+/ ϕ^/ ϕ),absentsuperscriptΩ2/ ^tensor-product¯𝜂¯𝜂^tensor-product¯𝜂12tr𝜒^¯𝜒12tr¯𝜒^𝜒/ italic-ϕ^tensor-product/ italic-ϕ\displaystyle=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\underline{\eta}+\underline{\eta}\widehat{\otimes}\underline{\eta}+\frac{1}{2}\mathrm{tr}\chi\widehat{\underline{\chi}}-\frac{1}{2}\mathrm{tr}\underline{\chi}\widehat{\chi}+\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi),
D¯(Ωtrχ)¯𝐷Ωtr𝜒\displaystyle\underline{D}(\Omega\mathrm{tr}\chi) =Ω2(2div/ η+2|η|2trχtrχ¯2K+2|/ ϕ|2),absentsuperscriptΩ22div/ 𝜂2superscript𝜂2tr𝜒tr¯𝜒2𝐾2superscript/ italic-ϕ2\displaystyle=\Omega^{2}(2\mbox{$\mathrm{div}\mkern-13.0mu/$ }\eta+2|\eta|^{2}-\mathrm{tr}\chi\mathrm{tr}\underline{\chi}-2K+2|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}),
D(Ωtrχ¯)𝐷Ωtr¯𝜒\displaystyle D(\Omega\mathrm{tr}\underline{\chi}) =Ω2(2div/ η¯+2|η¯|2trχtrχ¯2K+2|/ ϕ|2),absentsuperscriptΩ22div/ ¯𝜂2superscript¯𝜂2tr𝜒tr¯𝜒2𝐾2superscript/ italic-ϕ2\displaystyle=\Omega^{2}(2\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}+2|\underline{\eta}|^{2}-\mathrm{tr}\chi\mathrm{tr}\underline{\chi}-2K+2|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}),
Dη𝐷𝜂\displaystyle D\eta =(Ωχ)η¯(Ωβ+Lϕ/ ϕ),absentΩ𝜒¯𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle=(\Omega\chi)\cdot\underline{\eta}-(\Omega\beta+L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi),
D¯η¯¯𝐷¯𝜂\displaystyle\underline{D}\underline{\eta} =(Ωχ¯)η+(Ωβ¯L¯ϕ/ ϕ),absentΩ¯𝜒𝜂Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle=(\Omega\underline{\chi})\cdot\eta+(\Omega\underline{\beta}-\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi),
Dω¯𝐷¯𝜔\displaystyle D\underline{\omega} =Ω2(2(η,η¯)|η|2(ρ+16𝐑+L^ϕL¯^ϕ)),absentsuperscriptΩ22𝜂¯𝜂superscript𝜂2𝜌16𝐑^𝐿italic-ϕ^¯𝐿italic-ϕ\displaystyle=\Omega^{2}(2(\eta,\underline{\eta})-|\eta|^{2}-(\rho+\frac{1}{6}\mathbf{R}+\widehat{L}\phi\widehat{\underline{L}}\phi)),
D¯ω¯𝐷𝜔\displaystyle\underline{D}\omega =Ω2(2(η,η¯)|η¯|2(ρ+16𝐑+L^ϕL¯^ϕ)),absentsuperscriptΩ22𝜂¯𝜂superscript¯𝜂2𝜌16𝐑^𝐿italic-ϕ^¯𝐿italic-ϕ\displaystyle=\Omega^{2}(2(\eta,\underline{\eta})-|\underline{\eta}|^{2}-(\rho+\frac{1}{6}\mathbf{R}+\widehat{L}\phi\widehat{\underline{L}}\phi)),
K𝐾\displaystyle K =14trχtrχ¯+12(χ^,χ¯^)(ρ+16𝐑)+|/ ϕ|2,absent14tr𝜒tr¯𝜒12^𝜒^¯𝜒𝜌16𝐑superscript/ italic-ϕ2\displaystyle=-\frac{1}{4}\mathrm{tr}\chi\mathrm{tr}\underline{\chi}+\frac{1}{2}(\widehat{\chi},\widehat{\underline{\chi}})-(\rho+\frac{1}{6}\mathbf{R})+|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2},
div/ (Ωχ^)div/ Ω^𝜒\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi}) =12Ω2/ trχ+Ωχ^η¯+12Ωtrχη(ΩβLϕ/ ϕ),absent12superscriptΩ2/ trsuperscript𝜒Ω^𝜒¯𝜂12Ωtr𝜒𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle=\frac{1}{2}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}+\Omega\widehat{\chi}\cdot\underline{\eta}+\frac{1}{2}\Omega\mathrm{tr}\chi\eta-(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi),
div/ (Ωχ¯^)div/ Ω^¯𝜒\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}}) =12Ω2/ trχ¯+Ωχ¯^η+12Ωtrχ¯η¯+(Ωβ¯+L¯ϕ/ ϕ).absent12superscriptΩ2/ trsuperscript¯𝜒Ω^¯𝜒𝜂12Ωtr¯𝜒¯𝜂Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle=\frac{1}{2}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\underline{\chi}^{\prime}+\Omega\widehat{\underline{\chi}}\cdot\eta+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\underline{\eta}+(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi).

Recall the Einstein-scalar field system reads

{𝐑αβ=2αϕβϕ,gαβαβϕ=0.casessubscript𝐑𝛼𝛽2subscript𝛼italic-ϕsubscript𝛽italic-ϕotherwisesuperscript𝑔𝛼𝛽subscript𝛼subscript𝛽italic-ϕ0otherwise\begin{cases}\mathbf{R}_{\alpha\beta}=2\partial_{\alpha}\phi\partial_{\beta}\phi,\\ g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi=0\end{cases}.

Consequently, the spacetime scalar curvature 𝐑=2αϕαϕ𝐑2superscript𝛼italic-ϕsubscript𝛼italic-ϕ\mathbf{R}=2\partial^{\alpha}\phi\partial_{\alpha}\phi. The contracted second Bianchi identity will accordingly be the following inhomogeneous equation:

α𝐖αβγδ=[γ𝐑δ]β+16gβ[γδ]𝐑.\nabla^{\alpha}\mathbf{W}_{\alpha\beta\gamma\delta}=\nabla_{[\gamma}\mathbf{R}_{\delta]\beta}+\frac{1}{6}g_{\beta[\gamma}\nabla_{\delta]}\mathbf{R}.

This equation can be decomposed using the null frame (e1,e2,e3,e4)subscript𝑒1subscript𝑒2subscript𝑒3subscript𝑒4(e_{1},e_{2},e_{3},e_{4}) into components, which we call null Bianchi equations:121212See Proposition 1.2 of [9] in the vacuum case.

D¯^α12Ωtrχ¯α+2ω¯α+Ω{/ ^β(4η+ζ)^β+3χ^(ρ+16𝐑)+3χ^σ}^¯𝐷𝛼12Ωtr¯𝜒𝛼2¯𝜔𝛼Ω/ ^tensor-product𝛽4𝜂𝜁^tensor-product𝛽3^𝜒𝜌16𝐑3superscript^𝜒𝜎\displaystyle\widehat{\underline{D}}\alpha-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\alpha+2\underline{\omega}\alpha+\Omega\{-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\beta-(4\eta+\zeta)\widehat{\otimes}\beta+3\widehat{\chi}(\rho+\frac{1}{6}\mathbf{R})+3{}^{*}\widehat{\chi}\sigma\}
=\displaystyle= Ω(/ L^ϕ^/ ϕ/ ^/ ϕL^ϕ\displaystyle-\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{L}\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi\right.
12trχ/ ϕ^/ ϕχ^/ ϕ^/ ϕ+32χ^L^ϕL¯^ϕχ^|/ ϕ|2+12χ¯^(L^ϕ)2+ζ^/ ϕL^ϕ),\displaystyle\left.-\frac{1}{2}\mathrm{tr}\chi\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi-\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi+\frac{3}{2}\widehat{\chi}\widehat{L}\phi\widehat{\underline{L}}\phi-\widehat{\chi}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}+\frac{1}{2}\widehat{\underline{\chi}}(\widehat{L}\phi)^{2}+\zeta\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi\right),
Dβ+32ΩtrχβΩχ^βωβΩ{div/ α+(η¯+2ζ)α}𝐷𝛽32Ωtr𝜒𝛽Ω^𝜒𝛽𝜔𝛽Ωdiv/ 𝛼¯𝜂2𝜁𝛼\displaystyle D\beta+\frac{3}{2}\Omega\mathrm{tr}\chi\beta-\Omega\widehat{\chi}\cdot\beta-\omega\beta-\Omega\{\mbox{$\mathrm{div}\mkern-13.0mu/$ }\alpha+(\underline{\eta}+2\zeta)\cdot\alpha\}
=\displaystyle= Ω(/ ϕL^L^ϕL^ϕ/ L^ϕ+12trχ/ ϕL^ϕ+χ^/ ϕL^ϕΩ1ωL^ϕ/ ϕ(L^ϕ)2ζ),Ω/ italic-ϕ^𝐿^𝐿italic-ϕ^𝐿italic-ϕ/ ^𝐿italic-ϕ12tr𝜒/ italic-ϕ^𝐿italic-ϕ^𝜒/ italic-ϕ^𝐿italic-ϕsuperscriptΩ1𝜔^𝐿italic-ϕ/ italic-ϕsuperscript^𝐿italic-ϕ2𝜁\displaystyle\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\widehat{L}\phi-\widehat{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\widehat{L}\phi+\frac{1}{2}\mathrm{tr}\chi\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi+\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi-\Omega^{-1}\omega\widehat{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi-(\widehat{L}\phi)^{2}\zeta\right),
D¯β+12Ωtrχ¯βΩχ¯^β+ω¯βΩ{/ (ρ+16𝐑)+/ σ+3η(ρ+16𝐑)+3ησ+2χ^β¯}¯𝐷𝛽12Ωtr¯𝜒𝛽Ω^¯𝜒𝛽¯𝜔𝛽Ω/ 𝜌16𝐑superscript/ 𝜎3𝜂𝜌16𝐑3superscript𝜂𝜎2^𝜒¯𝛽\displaystyle\underline{D}\beta+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\beta-\Omega\widehat{\underline{\chi}}\cdot\beta+\underline{\omega}\beta-\Omega\{\mbox{$\nabla\mkern-13.0mu/$ }(\rho+\frac{1}{6}\mathbf{R})+{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\sigma+3\eta(\rho+\frac{1}{6}\mathbf{R})+3{}^{*}\eta\sigma+2\widehat{\chi}\cdot\underline{\beta}\}
=\displaystyle= Ω(/ ϕΔ/ ϕ/ L¯^ϕL^ϕ12trχL¯^ϕ/ ϕ+χ¯^/ ϕL^ϕ(ηζ)L¯^ϕL^ϕ+η|/ ϕ|2),Ω/ italic-ϕΔ/ italic-ϕ/ ^¯𝐿italic-ϕ^𝐿italic-ϕ12tr𝜒^¯𝐿italic-ϕ/ italic-ϕ^¯𝜒/ italic-ϕ^𝐿italic-ϕ𝜂𝜁^¯𝐿italic-ϕ^𝐿italic-ϕ𝜂superscript/ italic-ϕ2\displaystyle-\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\underline{L}}\phi\widehat{L}\phi-\frac{1}{2}\mathrm{tr}\chi\widehat{\underline{L}}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi-(\eta-\zeta)\widehat{\underline{L}}\phi\widehat{L}\phi+\eta|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\right),
D(ρ+16𝐑)+32Ωtrχ(ρ+16𝐑)Ω{div/ β+(2η¯+ζ,β)12(χ¯^,α)}𝐷𝜌16𝐑32Ωtr𝜒𝜌16𝐑Ωdiv/ 𝛽2¯𝜂𝜁𝛽12^¯𝜒𝛼\displaystyle D(\rho+\frac{1}{6}\mathbf{R})+\frac{3}{2}\Omega\mathrm{tr}\chi(\rho+\frac{1}{6}\mathbf{R})-\Omega\{\mbox{$\mathrm{div}\mkern-13.0mu/$ }\beta+(2\underline{\eta}+\zeta,\beta)-\frac{1}{2}(\widehat{\underline{\chi}},\alpha)\}
=\displaystyle= Ω(L^ϕΔϕ/ L^ϕ/ ϕ+χ^/ ϕ/ ϕ12trχ¯(L^ϕ)2ζ/ ϕL^ϕ),Ω^𝐿italic-ϕΔitalic-ϕ/ ^𝐿italic-ϕ/ italic-ϕ^𝜒/ italic-ϕ/ italic-ϕ12tr¯𝜒superscript^𝐿italic-ϕ2𝜁/ italic-ϕ^𝐿italic-ϕ\displaystyle-\Omega\left(\widehat{L}\phi\Delta\phi-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{L}\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi+\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi-\frac{1}{2}\mathrm{tr}\underline{\chi}(\widehat{L}\phi)^{2}-\zeta\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi\right),
Dσ+32Ωtrχσ+Ω{curl/ β+(2η¯+ζ,β)12χ¯^α}𝐷𝜎32Ωtr𝜒𝜎Ωcurl/ 𝛽2¯𝜂𝜁superscript𝛽12^¯𝜒𝛼\displaystyle D\sigma+\frac{3}{2}\Omega\mathrm{tr}\chi\sigma+\Omega\{\mbox{$\mathrm{curl}\mkern-13.0mu/$ }\beta+(2\underline{\eta}+\zeta,{}^{*}\beta)-\frac{1}{2}\widehat{\underline{\chi}}\wedge\alpha\}
=\displaystyle= Ω(/ L^ϕ/ ϕχ^/ ϕ/ ϕ+ζ/ ϕL^ϕ),Ω/ ^𝐿italic-ϕ/ italic-ϕ^𝜒/ italic-ϕ/ italic-ϕ𝜁/ italic-ϕ^𝐿italic-ϕ\displaystyle-\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{L}\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi-\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi+\zeta\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{L}\phi\right),
Dβ¯+12Ωtrχβ¯Ωχ^β¯+ωβ¯+Ω{/ (ρ+16𝐑)/ σ+3η¯(ρ+16𝐑)3η¯σ2χ¯^β}𝐷¯𝛽12Ωtr𝜒¯𝛽Ω^𝜒¯𝛽𝜔¯𝛽Ω/ 𝜌16𝐑superscript/ 𝜎3¯𝜂𝜌16𝐑3superscript¯𝜂𝜎2^¯𝜒𝛽\displaystyle D\underline{\beta}+\frac{1}{2}\Omega\mathrm{tr}\chi\underline{\beta}-\Omega\widehat{\chi}\cdot\underline{\beta}+\omega\underline{\beta}+\Omega\{\mbox{$\nabla\mkern-13.0mu/$ }(\rho+\frac{1}{6}\mathbf{R})-{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\sigma+3\underline{\eta}(\rho+\frac{1}{6}\mathbf{R})-3{}^{*}\underline{\eta}\sigma-2\widehat{\underline{\chi}}\cdot\beta\}
=\displaystyle= Ω(/ ϕΔ/ ϕ/ L^ϕL¯^ϕ12trχ¯L^ϕ/ ϕ+χ^/ ϕL¯^ϕ(η¯+ζ)L^ϕL¯^ϕ+η¯|/ ϕ|2),Ω/ italic-ϕΔ/ italic-ϕ/ ^𝐿italic-ϕ^¯𝐿italic-ϕ12tr¯𝜒^𝐿italic-ϕ/ italic-ϕ^𝜒/ italic-ϕ^¯𝐿italic-ϕ¯𝜂𝜁^𝐿italic-ϕ^¯𝐿italic-ϕ¯𝜂superscript/ italic-ϕ2\displaystyle\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{L}\phi\widehat{\underline{L}}\phi-\frac{1}{2}\mathrm{tr}\underline{\chi}\widehat{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi-(\underline{\eta}+\zeta)\widehat{L}\phi\widehat{\underline{L}}\phi+\underline{\eta}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\right),
D¯(ρ+16𝐑)+32Ωtrχ¯(ρ+16𝐑)+Ω{div/ β¯+(2ηζ,β¯)+12(χ^,α¯)}¯𝐷𝜌16𝐑32Ωtr¯𝜒𝜌16𝐑Ωdiv/ ¯𝛽2𝜂𝜁¯𝛽12^𝜒¯𝛼\displaystyle\underline{D}(\rho+\frac{1}{6}\mathbf{R})+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}(\rho+\frac{1}{6}\mathbf{R})+\Omega\{\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\beta}+(2\eta-\zeta,\underline{\beta})+\frac{1}{2}(\widehat{\chi},\underline{\alpha})\}
=\displaystyle= Ω(L¯^ϕΔϕ/ L¯^ϕ/ ϕ+χ¯^/ ϕ/ ϕ12trχ(L¯^ϕ)2+ζ/ ϕL¯^ϕ),Ω^¯𝐿italic-ϕΔitalic-ϕ/ ^¯𝐿italic-ϕ/ italic-ϕ^¯𝜒/ italic-ϕ/ italic-ϕ12tr𝜒superscript^¯𝐿italic-ϕ2𝜁/ italic-ϕ^¯𝐿italic-ϕ\displaystyle-\Omega\left(\widehat{\underline{L}}\phi\Delta\phi-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\underline{L}}\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi+\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi-\frac{1}{2}\mathrm{tr}\chi(\widehat{\underline{L}}\phi)^{2}+\zeta\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi\right),
D¯σ+32Ωtrχ¯σ+Ω{curl/ β¯+(2ηζ,β¯)+12χ^α¯}¯𝐷𝜎32Ωtr¯𝜒𝜎Ωcurl/ ¯𝛽2𝜂𝜁superscript¯𝛽12^𝜒¯𝛼\displaystyle\underline{D}\sigma+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}\sigma+\Omega\{\mbox{$\mathrm{curl}\mkern-13.0mu/$ }\underline{\beta}+(2\eta-\zeta,{}^{*}\underline{\beta})+\frac{1}{2}\widehat{\chi}\wedge\underline{\alpha}\}
=\displaystyle= Ω(/ L¯^ϕ/ ϕχ¯^/ ϕ/ ϕζ/ ϕL¯^ϕ),Ω/ ^¯𝐿italic-ϕ/ italic-ϕ^¯𝜒/ italic-ϕ/ italic-ϕ𝜁/ italic-ϕ^¯𝐿italic-ϕ\displaystyle\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\underline{L}}\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi-\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi-\zeta\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi\right),
D¯β¯+32Ωtrχ¯β¯Ωχ¯^β¯ω¯β¯+Ω{div/ α¯+(η2ζ)α¯}¯𝐷¯𝛽32Ωtr¯𝜒¯𝛽Ω^¯𝜒¯𝛽¯𝜔¯𝛽Ωdiv/ ¯𝛼𝜂2𝜁¯𝛼\displaystyle\underline{D}\underline{\beta}+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}\underline{\beta}-\Omega\widehat{\underline{\chi}}\cdot\underline{\beta}-\underline{\omega}\underline{\beta}+\Omega\{\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\alpha}+(\eta-2\zeta)\cdot\underline{\alpha}\}
=\displaystyle= Ω(/ ϕL¯^L¯^ϕL¯^ϕ/ L¯^ϕ+12trχ¯/ ϕL¯^ϕ+χ¯^/ ϕL¯^ϕΩ1ω¯L¯^ϕ/ ϕ+(L¯^ϕ)2ζ),Ω/ italic-ϕ^¯𝐿^¯𝐿italic-ϕ^¯𝐿italic-ϕ/ ^¯𝐿italic-ϕ12tr¯𝜒/ italic-ϕ^¯𝐿italic-ϕ^¯𝜒/ italic-ϕ^¯𝐿italic-ϕsuperscriptΩ1¯𝜔^¯𝐿italic-ϕ/ italic-ϕsuperscript^¯𝐿italic-ϕ2𝜁\displaystyle-\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\widehat{\underline{L}}\phi-\widehat{\underline{L}}\phi\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\underline{L}}\phi+\frac{1}{2}\mathrm{tr}\underline{\chi}\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi+\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi-\Omega^{-1}\underline{\omega}\widehat{\underline{L}}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+(\widehat{\underline{L}}\phi)^{2}\zeta\right),
D^α¯12Ωtrχα¯+2ωα¯+Ω{/ ^β¯+(4η¯ζ)^β¯+3χ¯^(ρ+16𝐑)3χ¯^σ}^𝐷¯𝛼12Ωtr𝜒¯𝛼2𝜔¯𝛼Ω/ ^tensor-product¯𝛽4¯𝜂𝜁^tensor-product¯𝛽3^¯𝜒𝜌16𝐑3superscript^¯𝜒𝜎\displaystyle\widehat{D}\underline{\alpha}-\frac{1}{2}\Omega\mathrm{tr}\chi\underline{\alpha}+2\omega\underline{\alpha}+\Omega\{\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\underline{\beta}+(4\underline{\eta}-\zeta)\widehat{\otimes}\underline{\beta}+3\widehat{\underline{\chi}}(\rho+\frac{1}{6}\mathbf{R})-3{}^{*}\widehat{\underline{\chi}}\sigma\}
=\displaystyle= Ω(/ L¯^ϕ^/ ϕ/ ^/ ϕL¯^ϕ\displaystyle-\Omega\left(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\underline{L}}\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi-\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi\right.
12trχ¯/ ϕ^/ ϕχ¯^/ ϕ^/ ϕ+32χ¯^L¯^ϕL^ϕχ¯^|/ ϕ|2+12χ^(L¯^ϕ)2ζ^/ ϕL¯^ϕ).\displaystyle\left.-\frac{1}{2}\mathrm{tr}\underline{\chi}\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi-\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi+\frac{3}{2}\widehat{\underline{\chi}}\widehat{\underline{L}}\phi\widehat{L}\phi-\widehat{\underline{\chi}}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}+\frac{1}{2}\widehat{\chi}(\widehat{\underline{L}}\phi)^{2}-\zeta\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\underline{L}}\phi\right).

As mentioned in the Introduction, we consider instead the following renormalized null Bianchi equations:

DK𝐷𝐾\displaystyle DK +ΩtrχK+div/ (ΩβLϕ/ ϕ)Ωtr𝜒𝐾div/ Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle+\Omega\mathrm{tr}\chi K+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
Ωχ^/ η¯+12Ωtrχdiv/ η¯+(ΩβLϕ/ ϕ)η¯Ωχ^η¯η¯+12Ωtrχ|η¯|2=0Ω^𝜒/ ¯𝜂12Ωtr𝜒div/ ¯𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ¯𝜂Ω^𝜒¯𝜂¯𝜂12Ωtr𝜒superscript¯𝜂20\displaystyle-\Omega\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\underline{\eta}+\frac{1}{2}\Omega\mathrm{tr}\chi\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}+(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\cdot\underline{\eta}-\Omega\widehat{\chi}\cdot\underline{\eta}\cdot\underline{\eta}+\frac{1}{2}\Omega\mathrm{tr}\chi|\underline{\eta}|^{2}=0
Dσˇ𝐷ˇ𝜎\displaystyle D\check{\sigma} +32Ωtrχσˇ+curl/ (ΩβLϕ/ ϕ)+12Ωχ^(η¯^η¯+/ ^η¯)32Ωtr𝜒ˇ𝜎curl/ Ω𝛽𝐿italic-ϕ/ italic-ϕ12Ω^𝜒¯𝜂^tensor-product¯𝜂/ ^tensor-product¯𝜂\displaystyle+\frac{3}{2}\Omega\mathrm{tr}\chi\check{\sigma}+\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\frac{1}{2}\Omega\widehat{\chi}\wedge(\underline{\eta}\widehat{\otimes}\underline{\eta}+\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\underline{\eta})
+η¯(ΩβLϕ/ ϕ)+2/ Lϕ/ ϕ=0¯𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ2/ 𝐿italic-ϕ/ italic-ϕ0\displaystyle+\underline{\eta}\wedge(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+2\mbox{$\nabla\mkern-13.0mu/$ }L\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi=0
D¯(ΩβLϕ/ ϕ)¯𝐷Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle\underline{D}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) +12Ωtrχ¯(ΩβLϕ/ ϕ)Ωχ¯^(ΩβLϕ/ ϕ)+Ω2/ KΩ2/ σˇ12Ωtr¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕΩ^¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕsuperscriptΩ2/ 𝐾superscriptΩ2superscript/ ˇ𝜎\displaystyle+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\Omega\widehat{\underline{\chi}}\cdot(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }K-\Omega^{2}{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma}
+3Ω2(ηKησˇ)12Ω2(/ (χ^,χ¯^)+/ (χ^χ¯^))32Ω2(η(χ^,χ¯^)+η(χ^χ¯^))3superscriptΩ2𝜂𝐾superscript𝜂ˇ𝜎12superscriptΩ2/ ^𝜒^¯𝜒superscript/ ^𝜒^¯𝜒32superscriptΩ2𝜂^𝜒^¯𝜒superscript𝜂^𝜒^¯𝜒\displaystyle+3\Omega^{2}(\eta K-{}^{*}\eta\check{\sigma})-\frac{1}{2}\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi},\widehat{\underline{\chi}})+{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi}\wedge\widehat{\underline{\chi}}))-\frac{3}{2}\Omega^{2}(\eta(\widehat{\chi},\widehat{\underline{\chi}})+{}^{*}\eta(\widehat{\chi}\wedge\widehat{\underline{\chi}}))
+14Ω2/ (trχtrχ¯)+34Ω2trχtrχ¯η2Ωχ^(Ωβ¯+L¯ϕ/ ϕ)14superscriptΩ2/ tr𝜒tr¯𝜒34superscriptΩ2tr𝜒tr¯𝜒𝜂2Ω^𝜒Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle+\frac{1}{4}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }(\mathrm{tr}\chi\mathrm{tr}\underline{\chi})+\frac{3}{4}\Omega^{2}\mathrm{tr}\chi\mathrm{tr}\underline{\chi}\eta-2\Omega\widehat{\chi}\cdot(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
=\displaystyle= 2Ω2Δ/ ϕ/ ϕ+Ω2/ |/ ϕ|22Ωχ^/ ϕL¯ϕ+ΩtrχL¯ϕ/ ϕ2superscriptΩ2Δ/ italic-ϕ/ italic-ϕsuperscriptΩ2/ superscript/ italic-ϕ22Ω^𝜒/ italic-ϕ¯𝐿italic-ϕΩtr𝜒¯𝐿italic-ϕ/ italic-ϕ\displaystyle-2\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}-2\Omega\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\underline{L}\phi+\Omega\mathrm{tr}\chi\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi
2Ω2η/ ϕ/ ϕ+2Ω2η|/ ϕ|2,2superscriptΩ2𝜂/ italic-ϕ/ italic-ϕ2superscriptΩ2𝜂superscript/ italic-ϕ2\displaystyle-2\Omega^{2}\eta\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+2\Omega^{2}\eta|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2},
D¯(K1|u|2)¯𝐷𝐾1superscript𝑢2\displaystyle\underline{D}(K-\frac{1}{|u|^{2}}) +32Ωtrχ¯(K1|u|2)+(Ωtrχ¯+2|u|)1|u|232Ωtr¯𝜒𝐾1superscript𝑢2Ωtr¯𝜒2𝑢1superscript𝑢2\displaystyle+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}(K-\frac{1}{|u|^{2}})+(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})\frac{1}{|u|^{2}}
=\displaystyle= div/ (Ωβ¯+L¯ϕ/ ϕ)+Ωχ¯^/ η+12Ωtrχ¯μdiv/ Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕΩ^¯𝜒/ 𝜂12Ωtr¯𝜒𝜇\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\Omega\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\eta+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\mu
+(Ωβ¯+L¯ϕ/ ϕ)η+Ωχ¯^ηη12Ωtrχ¯|η|2,Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ𝜂Ω^¯𝜒𝜂𝜂12Ωtr¯𝜒superscript𝜂2\displaystyle+(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\cdot\eta+\Omega\widehat{\underline{\chi}}\cdot\eta\cdot\eta-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}|\eta|^{2},
D¯σˇ¯𝐷ˇ𝜎\displaystyle\underline{D}\check{\sigma} +32Ωtrχσˇ+curl/ (Ωβ¯+L¯ϕ/ ϕ)32Ωtr𝜒ˇ𝜎curl/ Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle+\frac{3}{2}\Omega\mathrm{tr}\chi\check{\sigma}+\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
12Ωχ¯^(η^η+/ ^η)+η(Ωβ¯+L¯ϕ/ ϕ)2/ L¯ϕ/ ϕ=012Ω^¯𝜒𝜂^tensor-product𝜂/ ^tensor-product𝜂𝜂Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ2/ ¯𝐿italic-ϕ/ italic-ϕ0\displaystyle-\frac{1}{2}\Omega\widehat{\underline{\chi}}\wedge(\eta\widehat{\otimes}\eta+\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\eta)+\eta\wedge(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-2\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi=0
D(Ωβ¯+L¯ϕ/ ϕ)𝐷Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle D(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) +12Ωtrχ(Ωβ¯+L¯ϕ/ ϕ)Ωχ^(Ωβ¯+L¯ϕ/ ϕ)Ω2/ KΩ2/ σˇ12Ωtr𝜒Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕΩ^𝜒Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscriptΩ2/ 𝐾superscriptΩ2superscript/ ˇ𝜎\displaystyle+\frac{1}{2}\Omega\mathrm{tr}\chi(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\Omega\widehat{\chi}\cdot(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }K-\Omega^{2}{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma}
3Ω2(η¯K+η¯σˇ)+12Ω2(/ (χ^,χ¯^)/ (χ^χ¯^))+32Ω2(η¯(χ^,χ¯^)+η¯(χ^χ¯^))3superscriptΩ2¯𝜂𝐾superscript¯𝜂ˇ𝜎12superscriptΩ2/ ^𝜒^¯𝜒superscript/ ^𝜒^¯𝜒32superscriptΩ2¯𝜂^𝜒^¯𝜒superscript¯𝜂^𝜒^¯𝜒\displaystyle-3\Omega^{2}(\underline{\eta}K+{}^{*}\underline{\eta}\check{\sigma})+\frac{1}{2}\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi},\widehat{\underline{\chi}})-{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi}\wedge\widehat{\underline{\chi}}))+\frac{3}{2}\Omega^{2}(\underline{\eta}(\widehat{\chi},\widehat{\underline{\chi}})+{}^{*}\underline{\eta}(\widehat{\chi}\wedge\widehat{\underline{\chi}}))
14Ω2/ (trχtrχ¯)34Ω2trχtrχ¯η¯2Ωχ¯^(ΩβLϕ/ ϕ)14superscriptΩ2/ tr𝜒tr¯𝜒34superscriptΩ2tr𝜒tr¯𝜒¯𝜂2Ω^¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle-\frac{1}{4}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }(\mathrm{tr}\chi\mathrm{tr}\underline{\chi})-\frac{3}{4}\Omega^{2}\mathrm{tr}\chi\mathrm{tr}\underline{\chi}\underline{\eta}-2\Omega\widehat{\underline{\chi}}\cdot(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
=\displaystyle= 2Ω2Δ/ ϕ/ ϕΩ2/ |/ ϕ|2+2Ωχ¯^/ ϕLϕΩtrχ¯Lϕ/ ϕ2superscriptΩ2Δ/ italic-ϕ/ italic-ϕsuperscriptΩ2/ superscript/ italic-ϕ22Ω^¯𝜒/ italic-ϕ𝐿italic-ϕΩtr¯𝜒𝐿italic-ϕ/ italic-ϕ\displaystyle 2\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi-\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}+2\Omega\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi L\phi-\Omega\mathrm{tr}\underline{\chi}L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi
+2Ω2η¯/ ϕ/ ϕ2Ω2η¯|/ ϕ|2.2superscriptΩ2¯𝜂/ italic-ϕ/ italic-ϕ2superscriptΩ2¯𝜂superscript/ italic-ϕ2\displaystyle+2\Omega^{2}\underline{\eta}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi-2\Omega^{2}\underline{\eta}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}.

In the equation for D¯(K1|u|2)¯𝐷𝐾1superscript𝑢2\underline{D}(K-\frac{1}{|u|^{2}}) above, μ𝜇\mu is defined through

div/ η=K1|u|2μ.div/ 𝜂𝐾1superscript𝑢2𝜇\mbox{$\mathrm{div}\mkern-13.0mu/$ }\eta=K-\frac{1}{|u|^{2}}-\mu.

We also rewrite the wave equation gαβαβϕ=0superscript𝑔𝛼𝛽subscript𝛼subscript𝛽italic-ϕ0g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi=0 in the following form:

D¯Lϕ+12Ωtrχ¯Lϕ¯𝐷𝐿italic-ϕ12Ωtr¯𝜒𝐿italic-ϕ\displaystyle\underline{D}L\phi+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi =Ω2Δ/ ϕ+2Ω2(η,/ ϕ)12ΩtrχL¯ϕ,absentsuperscriptΩ2Δ/ italic-ϕ2superscriptΩ2𝜂/ italic-ϕ12Ωtr𝜒¯𝐿italic-ϕ\displaystyle=\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}(\eta,\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi,
D/ ϕ𝐷/ italic-ϕ\displaystyle D\mbox{$\nabla\mkern-13.0mu/$ }\phi =/ Lϕ,absent/ 𝐿italic-ϕ\displaystyle=\mbox{$\nabla\mkern-13.0mu/$ }L\phi,
DL¯ϕ+12ΩtrχL¯ϕ𝐷¯𝐿italic-ϕ12Ωtr𝜒¯𝐿italic-ϕ\displaystyle D\underline{L}\phi+\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi =Ω2Δ/ ϕ+2Ω2(η¯,/ ϕ)12Ωtrχ¯Lϕ,absentsuperscriptΩ2Δ/ italic-ϕ2superscriptΩ2¯𝜂/ italic-ϕ12Ωtr¯𝜒𝐿italic-ϕ\displaystyle=\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}(\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi,
D¯/ ϕ¯𝐷/ italic-ϕ\displaystyle\underline{D}\mbox{$\nabla\mkern-13.0mu/$ }\phi =/ L¯ϕ.absent/ ¯𝐿italic-ϕ\displaystyle=\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi.

Finally, the following elliptic-transport coupled systems are also needed:

{div/ η=K1|u|2μ,curl/ η=σ12χ^χ¯^,Dμ+Ωtrχμ=Ωtrχ1|u|2+div/ (2Ωχ^ηΩtrχη¯)+2/ Lϕ/ ϕ+2LϕΔ/ ϕ;casesdiv/ 𝜂𝐾1superscript𝑢2𝜇otherwisecurl/ 𝜂𝜎12^𝜒^¯𝜒otherwise𝐷𝜇Ωtr𝜒𝜇Ωtr𝜒1superscript𝑢2div/ 2Ω^𝜒𝜂Ωtr𝜒¯𝜂2/ 𝐿italic-ϕ/ italic-ϕ2𝐿italic-ϕΔ/ italic-ϕotherwise\displaystyle\begin{dcases}\mbox{$\mathrm{div}\mkern-13.0mu/$ }\eta=K-\frac{1}{|u|^{2}}-\mu,\\ \mbox{$\mathrm{curl}\mkern-13.0mu/$ }\eta=\sigma-\frac{1}{2}\widehat{\chi}\wedge\widehat{\underline{\chi}},\\ D\mu+\Omega\mathrm{tr}\chi\mu=-\Omega\mathrm{tr}\chi\frac{1}{|u|^{2}}+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(2\Omega\widehat{\chi}\cdot\eta-\Omega\mathrm{tr}\chi\underline{\eta})+2\mbox{$\nabla\mkern-13.0mu/$ }L\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi+2L\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi\end{dcases};
{div/ η¯=K1|u|2μ¯,curl/ η¯=σ+12χ^χ¯^,D¯μ¯+Ωtrχ¯μ¯=(Ωtrχ¯+2|u|)1|u|2+div/ (2Ωχ¯^η¯Ωtrχ¯η)+2/ L¯ϕ/ ϕ+2L¯ϕΔ/ ϕ;casesdiv/ ¯𝜂𝐾1superscript𝑢2¯𝜇otherwisecurl/ ¯𝜂𝜎12^𝜒^¯𝜒otherwise¯𝐷¯𝜇Ωtr¯𝜒¯𝜇Ωtr¯𝜒2𝑢1superscript𝑢2div/ 2Ω^¯𝜒¯𝜂Ωtr¯𝜒𝜂2/ ¯𝐿italic-ϕ/ italic-ϕ2¯𝐿italic-ϕΔ/ italic-ϕotherwise\displaystyle\begin{dcases}\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}=K-\frac{1}{|u|^{2}}-\underline{\mu},\\ \mbox{$\mathrm{curl}\mkern-13.0mu/$ }\underline{\eta}=-\sigma+\frac{1}{2}\widehat{\chi}\wedge\widehat{\underline{\chi}},\\ \underline{D}\underline{\mu}+\Omega\mathrm{tr}\underline{\chi}\underline{\mu}=-(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})\frac{1}{|u|^{2}}+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(2\Omega\widehat{\underline{\chi}}\cdot\underline{\eta}-\Omega\mathrm{tr}\underline{\chi}\eta)+2\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi+2\underline{L}\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi\end{dcases};
{Δ/ ω¯=ω¯/ +div/ (Ωβ¯+L¯ϕ/ ϕ),Dω¯/ +Ωtrχω¯/ +2Ωχ^/ / ω¯+2div/ (Ωχ^)/ ω¯12div/ (Ωtrχ(Ωβ¯+L¯ϕ/ ϕ))+/ (Ω2)(/ (ρ+16𝐑)+/ σ)+Δ/ (Ω2)(ρ+16𝐑)Δ/ (Ω2(2ηη¯|η|2))div/ (Ωχ^(Ωβ¯+L¯ϕ/ ϕ)2Ωχ¯^(ΩβLϕ/ ϕ)+3Ω2η¯(ρ+16𝐑)3Ω2η¯σ)=div/ {2Ω2/ ϕΔ/ ϕ+L¯ϕ/ Lϕ+Lϕ/ L¯ϕΩtrχ¯Lϕ/ ϕ+2Ωχ¯^/ ϕLϕ+2Ω2η¯/ ϕ/ ϕ+Ω2η¯|/ ϕ|2)}.\displaystyle\begin{dcases}\mbox{$\Delta\mkern-13.0mu/$ }\underline{\omega}&=\mbox{$\underline{\omega}\mkern-13.0mu/$ }+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi),\\ D\mbox{$\underline{\omega}\mkern-13.0mu/$ }&+\Omega\mathrm{tr}\chi\mbox{$\underline{\omega}\mkern-13.0mu/$ }+2\Omega\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}+2\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})\cdot\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}-\frac{1}{2}\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi))\\ &+\mbox{$\nabla\mkern-13.0mu/$ }(\Omega^{2})\cdot(\mbox{$\nabla\mkern-13.0mu/$ }(\rho+\frac{1}{6}\mathbf{R})+{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\sigma)+\mbox{$\Delta\mkern-13.0mu/$ }(\Omega^{2})(\rho+\frac{1}{6}\mathbf{R})-\mbox{$\Delta\mkern-13.0mu/$ }(\Omega^{2}(2\eta\cdot\underline{\eta}-|\eta|^{2}))\\ &-\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi}\cdot(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-2\Omega\widehat{\underline{\chi}}\cdot(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+3\Omega^{2}\underline{\eta}(\rho+\frac{1}{6}\mathbf{R})-3\Omega^{2}{}^{*}\underline{\eta}\sigma)\\ \phantom{\Delta}=&-\mbox{$\mathrm{div}\mkern-13.0mu/$ }\{2\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi+L\phi\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi-\Omega\mathrm{tr}\underline{\chi}L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\\ &+2\Omega\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi L\phi+2\Omega^{2}\underline{\eta}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\Omega^{2}\underline{\eta}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2})\}\end{dcases}.
Remark 2.1.

The elliptic-transport system for ω𝜔\omega is not needed because we renormalize the equations such that ω𝜔\omega does not appear. However, the system for ω¯¯𝜔\underline{\omega} is needed because it is crucial in estimating the top order derivatives of Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi}.

Before the end of this section, we list the commutation formulas which are used for the estimates of derivatives131313See Chapter 4 of [9] for the first group. The second group can be derived directly by the definition of curvature..

Lemma 2.1.

Given integer i𝑖i and tangential tensorfield ϕitalic-ϕ\phi. we have

[D,/ i]ϕ𝐷superscript/ 𝑖italic-ϕ\displaystyle[D,\mbox{$\nabla\mkern-13.0mu/$ }^{i}]\phi =j=1i/ j(Ωχ)/ ijϕ,absentsuperscriptsubscript𝑗1𝑖superscript/ 𝑗Ω𝜒superscript/ 𝑖𝑗italic-ϕ\displaystyle=\sum_{j=1}^{i}\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\Omega\chi)\cdot\mbox{$\nabla\mkern-13.0mu/$ }^{i-j}\phi,
[D¯,/ i]ϕ¯𝐷superscript/ 𝑖italic-ϕ\displaystyle[\underline{D},\mbox{$\nabla\mkern-13.0mu/$ }^{i}]\phi =j=1i/ j(Ωχ¯)/ ijϕ,absentsuperscriptsubscript𝑗1𝑖superscript/ 𝑗Ω¯𝜒superscript/ 𝑖𝑗italic-ϕ\displaystyle=\sum_{j=1}^{i}\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\Omega\underline{\chi})\cdot\mbox{$\nabla\mkern-13.0mu/$ }^{i-j}\phi,

and

[𝒟,/ i]ϕ𝒟superscript/ 𝑖italic-ϕ\displaystyle[\mathcal{D},\mbox{$\nabla\mkern-13.0mu/$ }^{i}]\phi =j=1i/ j1K/ ijϕ,absentsuperscriptsubscript𝑗1𝑖superscript/ 𝑗1𝐾superscript/ 𝑖𝑗italic-ϕ\displaystyle=\sum_{j=1}^{i}\mbox{$\nabla\mkern-13.0mu/$ }^{j-1}K\cdot\mbox{$\nabla\mkern-13.0mu/$ }^{i-j}\phi,
[𝒟,/ i]ϕ\displaystyle[^{*}\mathcal{D},\mbox{$\nabla\mkern-13.0mu/$ }^{i}]\phi =j=1i/ j1K/ ijϕ.absentsuperscriptsubscript𝑗1𝑖superscript/ 𝑗1𝐾superscript/ 𝑖𝑗italic-ϕ\displaystyle=\sum_{j=1}^{i}\mbox{$\nabla\mkern-13.0mu/$ }^{j-1}K\cdot\mbox{$\nabla\mkern-13.0mu/$ }^{i-j}\phi.

Here we use “\cdot” to represent an arbitrary contraction with the coefficients by g/g\mkern-9.0mu/ or ϵ/\epsilon\mkern-9.0mu/. In addition, if ϕitalic-ϕ\phi is a function, then when i=1𝑖1i=1, all commutators above are zero; when i2𝑖2i\geq 2, all i𝑖i’s are replaced by i1𝑖1i-1’s in above formulas.

3. Statement of the existence theorem

3.1. Formulation of the problem

We formula the problem we are going to consider again. We consider a double characteristic initial value problem of the Einstein-scalar field equations, where the initial data is given on two null cone, Cu0subscript𝐶subscript𝑢0C_{u_{0}} which is outgoing, and C¯0subscript¯𝐶0\underline{C}_{0} which is incoming, intersecting on a sphere S0,u0subscript𝑆0subscript𝑢0S_{0,u_{0}}. The data on C¯0subscript¯𝐶0\underline{C}_{0} is spherically symmetric. The restriction of the optical function u𝑢u on C¯0subscript¯𝐶0\underline{C}_{0} is chosen such that the level sets of u𝑢u on C¯0subscript¯𝐶0\underline{C}_{0} are the spherical sections of the symmetry, and in addition u=r𝑢𝑟u=-r where r𝑟r is the area radius of the corresponding section.

The data on C¯0subscript¯𝐶0\underline{C}_{0} then consists of ϕitalic-ϕ\phi up to a constant (or L¯ϕ¯𝐿italic-ϕ\underline{L}\phi) and ΩΩ\Omega, where ϕitalic-ϕ\phi is the scalar field and L¯=u¯𝐿subscript𝑢\underline{L}=-\partial_{u} on C¯0subscript¯𝐶0\underline{C}_{0} (χ¯^^¯𝜒\widehat{\underline{\chi}} vanishes because of spherical symmetry). Now denote

ψ=ψ(u)=|u|L¯ϕ|C¯0,h=h(u)=|u|trχ2|C¯0,Ω0=Ω0(u)=Ω|C¯0.formulae-sequence𝜓𝜓𝑢evaluated-at𝑢¯𝐿italic-ϕsubscript¯𝐶0𝑢evaluated-at𝑢trsuperscript𝜒2subscript¯𝐶0subscriptΩ0subscriptΩ0𝑢evaluated-atΩsubscript¯𝐶0\psi=\psi(u)=|u|\underline{L}\phi\Big{|}_{\underline{C}_{0}},\ h=h(u)=\frac{|u|\mathrm{tr}\chi^{\prime}}{2}\Big{|}_{\underline{C}_{0}},\ \Omega_{0}=\Omega_{0}(u)=\Omega\Big{|}_{\underline{C}_{0}}.

We require that these three functions are smooth for u[u0,0)𝑢subscript𝑢00u\in[u_{0},0) and 0<h1010<h\leq 1. In fact, from the Raychaudhuri equation along C¯0subscript¯𝐶0\underline{C}_{0}, and Ωtrχ¯=2|u|Ωtr¯𝜒2𝑢\Omega\mathrm{tr}\underline{\chi}=-\frac{2}{|u|} which follows by setting u=r𝑢𝑟u=-r, Ω0subscriptΩ0\Omega_{0} is determined by ψ𝜓\psi through

D¯logΩ0=(ω¯|C¯0=)12ψ2|u|,\displaystyle\underline{D}\log\Omega_{0}=\left(\underline{\omega}\Big{|}_{\underline{C}_{0}}=\right)-\frac{1}{2}\frac{\psi^{2}}{|u|}, (3.1)

and from the null structure equation for D¯(Ωtrχ)¯𝐷Ωtr𝜒\underline{D}(\Omega\mathrm{tr}\chi), hh is determined by Ω0subscriptΩ0\Omega_{0} through

D¯(Ω02h)=1|u|Ω02(h1).¯𝐷superscriptsubscriptΩ021𝑢superscriptsubscriptΩ021\displaystyle\underline{D}(\Omega_{0}^{2}h)=\frac{1}{|u|}\Omega_{0}^{2}(h-1). (3.2)

The function Ω0subscriptΩ0\Omega_{0} will play a very important role in the whole paper. It has two important properties stated in Lemma 1.1 which will be used frequently: Ω0subscriptΩ0\Omega_{0} is monotonically decreasing, and if the vertex of C¯0subscript¯𝐶0\underline{C}_{0} is singular, then Ω00subscriptΩ00\Omega_{0}\to 0 as u0𝑢superscript0u\to 0^{-}.

3.2. Norms

We first introduce the following scale invariant norms relative to the double null foliation (for 2p2𝑝2\leq p\leq\infty and q=1,2𝑞12q=1,2):

ξ𝕃p(u¯,u)=subscriptnorm𝜉superscript𝕃𝑝¯𝑢𝑢absent\displaystyle\|\xi\|_{\mathbb{L}^{p}(\underline{u},u)}= (Su¯,u|u|2|ξ|pdμg/)1p,\displaystyle\left(\int_{S_{\underline{u},u}}|u|^{-2}|\xi|^{p}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\right)^{\frac{1}{p}},
ξn(u¯,u)=subscriptnorm𝜉superscript𝑛¯𝑢𝑢absent\displaystyle\|\xi\|_{\mathbb{H}^{n}(\underline{u},u)}= i=0n(|u|/ )iξ𝕃2(u¯,u),superscriptsubscript𝑖0𝑛subscriptnormsuperscript𝑢/ 𝑖𝜉superscript𝕃2¯𝑢𝑢\displaystyle\sum_{i=0}^{n}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}\xi\|_{\mathbb{L}^{2}(\underline{u},u)},
ξ𝕃[u1,u2]qn(u¯)=subscriptnorm𝜉subscriptsuperscript𝕃𝑞subscript𝑢1subscript𝑢2superscript𝑛¯𝑢absent\displaystyle\|\xi\|_{\mathbb{L}^{q}_{[u_{1},u_{2}]}\mathbb{H}^{n}(\underline{u})}= (u1u2|u|1ξn(u¯,u)qdu)1q,superscriptsuperscriptsubscriptsubscript𝑢1subscript𝑢2superscript𝑢1superscriptsubscriptnorm𝜉superscript𝑛¯𝑢superscript𝑢𝑞differential-dsuperscript𝑢1𝑞\displaystyle\left(\int_{u_{1}}^{u_{2}}|u|^{-1}\|\xi\|_{\mathbb{H}^{n}(\underline{u},u^{\prime})}^{q}\mathrm{d}u^{\prime}\right)^{\frac{1}{q}},
ξ𝕃u¯qn(u)=subscriptnorm𝜉subscriptsuperscript𝕃𝑞¯𝑢superscript𝑛𝑢absent\displaystyle\|\xi\|_{\mathbb{L}^{q}_{\underline{u}}\mathbb{H}^{n}(u)}= (0δδ1ξn(u¯,u)qdu¯)1q.superscriptsuperscriptsubscript0𝛿superscript𝛿1superscriptsubscriptnorm𝜉superscript𝑛superscript¯𝑢𝑢𝑞differential-dsuperscript¯𝑢1𝑞\displaystyle\left(\int_{0}^{\delta}\delta^{-1}\|\xi\|_{\mathbb{H}^{n}(\underline{u}^{\prime},u)}^{q}\mathrm{d}\underline{u}^{\prime}\right)^{\frac{1}{q}}.

In addition, we define

ξ𝕃u¯𝕃[u1,u2]qn=subscriptnorm𝜉subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃𝑞subscript𝑢1subscript𝑢2superscript𝑛absent\displaystyle\|\xi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{q}_{[u_{1},u_{2}]}\mathbb{H}^{n}}= supu¯[0,δ](u1u2|u|1ξn(u¯,u)qdu)1q,subscriptsupremumsuperscript¯𝑢0𝛿superscriptsuperscriptsubscriptsubscript𝑢1subscript𝑢2superscript𝑢1superscriptsubscriptnorm𝜉superscript𝑛superscript¯𝑢superscript𝑢𝑞differential-dsuperscript𝑢1𝑞\displaystyle\sup_{\underline{u}^{\prime}\in[0,\delta]}\left(\int_{u_{1}}^{u_{2}}|u|^{-1}\|\xi\|_{\mathbb{H}^{n}(\underline{u}^{\prime},u^{\prime})}^{q}\mathrm{d}u^{\prime}\right)^{\frac{1}{q}},
ξ𝕃[u1,u2]𝕃u¯qn=subscriptnorm𝜉subscriptsuperscript𝕃subscript𝑢1subscript𝑢2subscriptsuperscript𝕃𝑞¯𝑢superscript𝑛absent\displaystyle\|\xi\|_{\mathbb{L}^{\infty}_{[u_{1},u_{2}]}\mathbb{L}^{q}_{\underline{u}}\mathbb{H}^{n}}= supu[u1,u2](0δξn(u¯,u)qdu¯)1q.subscriptsupremumsuperscript𝑢subscript𝑢1subscript𝑢2superscriptsuperscriptsubscript0𝛿superscriptsubscriptnorm𝜉superscript𝑛superscript¯𝑢superscript𝑢𝑞differential-dsuperscript¯𝑢1𝑞\displaystyle\sup_{u^{\prime}\in[u_{1},u_{2}]}\left(\int_{0}^{\delta}\|\xi\|_{\mathbb{H}^{n}(\underline{u}^{\prime},u^{\prime})}^{q}\mathrm{d}\underline{u}^{\prime}\right)^{\frac{1}{q}}.

We are going to define the norms of various geometric quantities which will be used in the proof of the existence theorem. Set a function

=(δ,u0,u1)1,𝛿subscript𝑢0subscript𝑢11\displaystyle\mathscr{F}=\mathscr{F}(\delta,u_{0},u_{1})\geq 1,

for some δ>0𝛿0\delta>0 and u1(u0,0)subscript𝑢1subscript𝑢00u_{1}\in(u_{0},0) and define

=(δ,u0,u1):=𝛿subscript𝑢0subscript𝑢1assignabsent\displaystyle\mathscr{E}=\mathscr{E}(\delta,u_{0},u_{1}):= max{1,1𝒜1|u0|(|u0|/ )Lϕ𝕃[0,δ]24(u0)|log|u1||u0||12}1,1superscript1superscript𝒜1evaluated-atsubscript𝑢0subscript𝑢0/ 𝐿italic-ϕsubscriptsuperscript𝕃20𝛿superscript4subscript𝑢0superscriptsubscript𝑢1subscript𝑢0121\displaystyle\max\left\{1,\mathscr{F}^{-1}\mathcal{A}^{-1}\||u_{0}|(|u_{0}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{[0,\delta]}\mathbb{H}^{4}(u_{0})}\left|\log\frac{|u_{1}|}{|u_{0}|}\right|^{\frac{1}{2}}\right\}\geq 1,
𝒲=𝒲(u0,u1):=𝒲𝒲subscript𝑢0subscript𝑢1assignabsent\displaystyle\mathscr{W}=\mathscr{W}(u_{0},u_{1}):= max{1,|logΩ0(u1)Ω0(u0)|}11subscriptΩ0subscript𝑢1subscriptΩ0subscript𝑢01\displaystyle\max\left\{1,\left|\log\frac{\Omega_{0}(u_{1})}{\Omega_{0}(u_{0})}\right|\right\}\geq 1

where

𝒜=𝒜(δ,u0,u1)𝒜𝒜𝛿subscript𝑢0subscript𝑢1\displaystyle\mathcal{A}=\mathcal{A}(\delta,u_{0},u_{1})

is the bound of the initial data which will be defined precisely in the statement of Theorem 3.1. Now we define

𝒪(u¯,u;u0,u1)𝒪¯𝑢𝑢subscript𝑢0subscript𝑢1\displaystyle\mathcal{O}(\underline{u},u;u_{0},u_{1})
=\displaystyle= [δ1|u|2Ω022trχ2h|u|4(u¯,u)]12superscriptdelimited-[]superscript𝛿1superscript𝑢2superscriptsubscriptΩ02superscript2subscriptnormtrsuperscript𝜒2𝑢superscript4¯𝑢𝑢12\displaystyle\left[\delta^{-1}|u|^{2}\Omega_{0}^{2}\mathscr{F}^{-2}\left\|\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|}\right\|_{\mathbb{H}^{4}(\underline{u},u)}\right]^{\frac{1}{2}}
+δ1|u|21Ωtrχ¯+2|u|,Ωχ¯^4(u¯,u)\displaystyle+\delta^{-1}|u|^{2}\mathscr{F}^{-1}\left\|\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|},\Omega\widehat{\underline{\chi}}\right\|_{\mathbb{H}^{4}(\underline{u},u)}
+|u|1(Ωχ^4(u¯,u)+𝒲12ω4(u¯,u))𝑢superscript1subscriptnormΩ^𝜒superscript4¯𝑢𝑢superscript𝒲12subscriptnorm𝜔superscript4¯𝑢𝑢\displaystyle+|u|\mathscr{F}^{-1}\left(\|\Omega\widehat{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}+\mathscr{W}^{-\frac{1}{2}}\|\omega\|_{\mathbb{H}^{4}(\underline{u},u)}\right)
+δ1|u|211(η4(u¯,u)+𝒲12η¯4(u¯,u)).superscript𝛿1superscript𝑢2superscript1superscript1subscriptnorm𝜂superscript4¯𝑢𝑢superscript𝒲12subscriptnorm¯𝜂superscript4¯𝑢𝑢\displaystyle+\delta^{-1}|u|^{2}\mathscr{F}^{-1}\mathscr{E}^{-1}\left(\|\eta\|_{\mathbb{H}^{4}(\underline{u},u)}+\mathscr{W}^{-\frac{1}{2}}\|\underline{\eta}\|_{\mathbb{H}^{4}(\underline{u},u)}\right).
(u¯,u;u0,u1)=¯𝑢𝑢subscript𝑢0subscript𝑢1absent\displaystyle\mathcal{E}(\underline{u},u;u_{0},u_{1})= [δ11|u|14|u|34(|u|L¯ϕψ)𝕃[u0,u1]24(u¯)\displaystyle\left[\delta^{-1}\mathscr{F}^{-1}|u|^{\frac{1}{4}}\||u|^{\frac{3}{4}}(|u|\underline{L}\phi-\psi)\|_{\mathbb{L}_{[u_{0},u_{1}]}^{2}\mathbb{H}^{4}(\underline{u})}\right.
+1|u|Lϕφ4(u¯,u)superscript1subscriptnorm𝑢𝐿italic-ϕ𝜑superscript4¯𝑢𝑢\displaystyle+\mathscr{F}^{-1}\||u|L\phi-\varphi\|_{\mathbb{H}^{4}(\underline{u},u)}
+δ1|u|211/ ϕ4(u¯,u)].\displaystyle\left.+\delta^{-1}|u|^{2}\mathscr{F}^{-1}\mathscr{E}^{-1}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}\right].
𝒪~(u¯,u;u0;u1)=~𝒪¯𝑢𝑢subscript𝑢0subscript𝑢1absent\displaystyle\widetilde{\mathcal{O}}(\underline{u},u;u_{0};u_{1})=
[δ1|u|2Ω0221(|u|/ )trχ4(u¯,u)]12superscriptdelimited-[]superscript𝛿1superscript𝑢2superscriptsubscriptΩ02superscript2superscript1subscriptnorm𝑢/ trsuperscript𝜒superscript4¯𝑢𝑢12\displaystyle\left[\delta^{-1}|u|^{2}\Omega_{0}^{2}\mathscr{F}^{-2}\mathscr{E}^{-1}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mathrm{tr}\chi^{\prime}\|_{\mathbb{H}^{4}(\underline{u},u)}\right]^{\frac{1}{2}}
+δ1|u|211𝒲12(|u|/ )(Ωtrχ¯)4(u¯,u)superscript𝛿1superscript𝑢2superscript1superscript1superscript𝒲12subscriptnorm𝑢/ Ωtr¯𝜒superscript4¯𝑢𝑢\displaystyle+\delta^{-1}|u|^{2}\mathscr{F}^{-1}\mathscr{E}^{-1}\mathscr{W}^{-\frac{1}{2}}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{\mathbb{H}^{4}(\underline{u},u)}
+|u|11Ωχ^𝕃u¯25(u)𝑢superscript1superscript1subscriptnormΩ^𝜒subscriptsuperscript𝕃2¯𝑢superscript5𝑢\displaystyle+|u|\mathscr{F}^{-1}\mathscr{E}^{-1}\|\Omega\widehat{\chi}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}
+δ12|u|32Ω011𝒲12η¯𝕃u¯25(u)superscript𝛿12superscript𝑢32subscriptΩ0superscript1superscript1superscript𝒲12subscriptnorm¯𝜂subscriptsuperscript𝕃2¯𝑢superscript5𝑢\displaystyle+\delta^{-\frac{1}{2}}|u|^{\frac{3}{2}}\Omega_{0}\mathscr{F}^{-1}\mathscr{E}^{-1}\mathscr{W}^{-\frac{1}{2}}\|\underline{\eta}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}
+δ12|u|32Ω011η𝕃u¯25(u)superscript𝛿12superscript𝑢32subscriptΩ0superscript1superscript1subscriptnorm𝜂superscriptsubscript𝕃¯𝑢2superscript5𝑢\displaystyle+\delta^{-\frac{1}{2}}|u|^{\frac{3}{2}}\Omega_{0}\mathscr{F}^{-1}\mathscr{E}^{-1}\|\eta\|_{\mathbb{L}_{\underline{u}}^{2}\mathbb{H}^{5}(u)}
+δ1211Ω0|u|32η𝕃[u0,u]25(u¯)superscript𝛿12superscript1superscript1subscriptnormsubscriptΩ0superscript𝑢32𝜂superscriptsubscript𝕃subscript𝑢0𝑢2superscript5¯𝑢\displaystyle+\delta^{-\frac{1}{2}}\mathscr{F}^{-1}\mathscr{E}^{-1}\|\Omega_{0}|u|^{\frac{3}{2}}\eta\|_{\mathbb{L}_{[u_{0},u]}^{2}\mathbb{H}^{5}(\underline{u})}
+δ1|u|12Ω011𝒲12Ω01|u|32(Ωχ¯^)𝕃[u0,u]25(u¯)},\displaystyle+\delta^{-1}|u|^{\frac{1}{2}}\Omega_{0}\mathscr{F}^{-1}\mathscr{E}^{-1}\mathscr{W}^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{3}{2}}(\Omega\widehat{\underline{\chi}})\|_{\mathbb{L}_{[u_{0},u]}^{2}\mathbb{H}^{5}(\underline{u})}\},
~(u;u0;u1)~𝑢subscript𝑢0subscript𝑢1\displaystyle\widetilde{\mathcal{E}}(u;u_{0};u_{1})
=\displaystyle= 11(|u|(|u|/ )Lϕ𝕃u¯24(u)+δ12|u|32Ω0(|u|/ )/ ϕ𝕃u¯𝕃[u0,u]24)superscript1superscript1subscriptnorm𝑢𝑢/ 𝐿italic-ϕsuperscriptsubscript𝕃¯𝑢2superscript4𝑢superscript𝛿12subscriptnormsuperscript𝑢32subscriptΩ0𝑢/ / italic-ϕsubscriptsuperscript𝕃¯𝑢superscriptsubscript𝕃subscript𝑢0𝑢2superscript4\displaystyle\mathscr{F}^{-1}\mathscr{E}^{-1}\left(\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}_{\underline{u}}^{2}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\||u|^{\frac{3}{2}}\Omega_{0}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}_{[u_{0},u]}^{2}\mathbb{H}^{4}}\right)
+Ω0δ1211(|u|32(|u|/ )/ ϕ𝕃u¯24(u)+δ12Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24(δ)),subscriptΩ0superscript𝛿12superscript1superscript1subscriptnormsuperscript𝑢32𝑢/ / italic-ϕsuperscriptsubscript𝕃¯𝑢2superscript4𝑢superscript𝛿12subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢superscriptsubscript𝕃subscript𝑢0𝑢2superscript4𝛿\displaystyle+\Omega_{0}\delta^{-\frac{1}{2}}\mathscr{F}^{-1}\mathscr{E}^{-1}\left(\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}_{\underline{u}}^{2}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}_{[u_{0},u]}^{2}\mathbb{H}^{4}(\delta)}\right),
(u;u0;u1)𝑢subscript𝑢0subscript𝑢1\displaystyle\mathcal{R}(u;u_{0};u_{1})
=\displaystyle= 11(|u|2(ΩβLϕ/ ϕ)𝕃u¯24(u)+δ12|u|52Ω0(K1|u|2,σˇ)𝕃u¯𝕃[u0,u]24)superscript1superscript1subscriptnormsuperscript𝑢2Ω𝛽𝐿italic-ϕ/ italic-ϕsuperscriptsubscript𝕃¯𝑢2superscript4𝑢superscript𝛿12subscriptnormsuperscript𝑢52subscriptΩ0𝐾1superscript𝑢2ˇ𝜎subscriptsuperscript𝕃¯𝑢superscriptsubscript𝕃subscript𝑢0𝑢2superscript4\displaystyle\mathscr{F}^{-1}\mathscr{E}^{-1}\left(\||u|^{2}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}_{\underline{u}}^{2}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\left\||u|^{\frac{5}{2}}\Omega_{0}(K-\frac{1}{|u|^{2}},\check{\sigma})\right\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}_{[u_{0},u]}^{2}\mathbb{H}^{4}}\right)
+[Ω0δ1321\displaystyle+\left[\Omega_{0}\delta^{-1}\mathscr{F}^{-\frac{3}{2}}\mathscr{E}^{-1}\right.
×(|u|3(K1|u|2,σˇ)𝕃u¯24(u)+δ12Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)𝕃u¯𝕃[u0,u]24)]23.\displaystyle\ \ \ \left.\times\left(\||u|^{3}(K-\frac{1}{|u|^{2}},\check{\sigma})\|_{\mathbb{L}_{\underline{u}}^{2}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}_{[u_{0},u]}^{2}\mathbb{H}^{4}}\right)\right]^{\frac{2}{3}}.
Remark 3.1.

Some remarks should be made about these norms. First, in the definitions above, Ω0subscriptΩ0\Omega_{0} appearing outside the norm symbol “\|\cdot\|” means its value at u𝑢u, i.e., Ω0=Ω0(u)subscriptΩ0subscriptΩ0𝑢\Omega_{0}=\Omega_{0}(u). In the rest of the paper, Ω0subscriptΩ0\Omega_{0} outside “\|\cdot\|” should be understood in the same way. On the other hand, if Ω0subscriptΩ0\Omega_{0} or Ω01superscriptsubscriptΩ01\Omega_{0}^{-1} appears inside the norm 𝕃[u0,u]2\|\cdot\|_{\mathbb{L}^{2}_{[u_{0},u]}}, then it is a factor of the integrand. All factors inside the norm 𝕃[u0,u]2\|\cdot\|_{\mathbb{L}^{2}_{[u_{0},u]}} cannot be taken out directly. Second, the norm

(|u|/ )(Ωtrχ)4(u¯,u)subscriptnorm𝑢/ Ωtr𝜒superscript4¯𝑢𝑢\displaystyle\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{\mathbb{H}^{4}(\underline{u},u)}

is a norm about up to the \engordnumber5 order derivatives of Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} except itself. For some of the geometric quantities, the derivatives obey better estimates as compared to the quantities themselves.

Finally, we define

𝒪=𝒪(u0,u1)𝒪𝒪subscript𝑢0subscript𝑢1\displaystyle\mathcal{O}=\mathcal{O}(u_{0},u_{1}) =sup0u¯δu0uu1𝒪(u¯,u;u0,u1),absentsubscriptsupremumFRACOP0¯𝑢𝛿subscript𝑢0𝑢subscript𝑢1𝒪¯𝑢𝑢subscript𝑢0subscript𝑢1\displaystyle=\sup_{0\leq\underline{u}\leq\delta\atop u_{0}\leq u\leq u_{1}}\mathcal{O}(\underline{u},u;u_{0},u_{1}),
=(u0,u1)subscript𝑢0subscript𝑢1\displaystyle\mathcal{E}=\mathcal{E}(u_{0},u_{1}) =sup0u¯δu0uu1(u¯,u;u0,u1),absentsubscriptsupremumFRACOP0¯𝑢𝛿subscript𝑢0𝑢subscript𝑢1¯𝑢𝑢subscript𝑢0subscript𝑢1\displaystyle=\sup_{0\leq\underline{u}\leq\delta\atop u_{0}\leq u\leq u_{1}}\mathcal{E}(\underline{u},u;u_{0},u_{1}),
𝒪~=𝒪~(u0,u1)~𝒪~𝒪subscript𝑢0subscript𝑢1\displaystyle\widetilde{\mathcal{O}}=\widetilde{\mathcal{O}}(u_{0},u_{1}) =sup0u¯δu0uu1𝒪~(u¯,u;u0,u1),absentsubscriptsupremumFRACOP0¯𝑢𝛿subscript𝑢0𝑢subscript𝑢1~𝒪¯𝑢𝑢subscript𝑢0subscript𝑢1\displaystyle=\sup_{0\leq\underline{u}\leq\delta\atop u_{0}\leq u\leq u_{1}}\widetilde{\mathcal{O}}(\underline{u},u;u_{0},u_{1}),
~=~(u0,u1)~~subscript𝑢0subscript𝑢1\displaystyle\widetilde{\mathcal{E}}=\widetilde{\mathcal{E}}(u_{0},u_{1}) =supu0uu1~(u;u0,u1),absentsubscriptsupremumsubscript𝑢0𝑢subscript𝑢1~𝑢subscript𝑢0subscript𝑢1\displaystyle=\sup_{u_{0}\leq u\leq u_{1}}\widetilde{\mathcal{E}}(u;u_{0},u_{1}),
=(u0,u1)subscript𝑢0subscript𝑢1\displaystyle\mathcal{R}=\mathcal{R}(u_{0},u_{1}) =supu0uu1(u;u0,u1).absentsubscriptsupremumsubscript𝑢0𝑢subscript𝑢1𝑢subscript𝑢0subscript𝑢1\displaystyle=\sup_{u_{0}\leq u\leq u_{1}}\mathcal{R}(u;u_{0},u_{1}).
Theorem 3.1 (Existence Theorem).

There exists a universal constant C01subscript𝐶01C_{0}\geq 1 such that the following statement is true. Let CC0𝐶subscript𝐶0C\geq C_{0}, δ>0𝛿0\delta>0, u0subscript𝑢0u_{0} and u1subscript𝑢1u_{1} be four numbers such that u0<u1<0subscript𝑢0subscript𝑢10u_{0}<u_{1}<0. Suppose that the smooth initial data given on Cu0C¯0subscript𝐶subscript𝑢0subscript¯𝐶0C_{u_{0}}\bigcup\underline{C}_{0} is as described above. The data on C¯0subscript¯𝐶0\underline{C}_{0} is spherically symmetric and Ω0(u0)1subscriptΩ0subscript𝑢01\Omega_{0}(u_{0})\leq 1 and the data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} satisfies

𝒜:=max{1,supu0uu1(1|φ(u)|),1|u0|sup0u¯δ(Ωχ^7(u¯,u0)+ω5(u¯,u0)+Lϕ5(u¯,u0))}<+.assign𝒜1subscriptsupremumsubscript𝑢0𝑢subscript𝑢1superscript1𝜑𝑢superscript1subscript𝑢0subscriptsupremum0¯𝑢𝛿subscriptdelimited-∥∥Ω^𝜒superscript7¯𝑢subscript𝑢0subscriptdelimited-∥∥𝜔superscript5¯𝑢subscript𝑢0subscriptdelimited-∥∥𝐿italic-ϕsuperscript5¯𝑢subscript𝑢0\begin{split}\mathcal{A}:=&\max\left\{1,\sup_{u_{0}\leq u\leq u_{1}}\left(\mathscr{F}^{-1}|\varphi(u)|\right),\right.\\ &\left.\mathscr{F}^{-1}|u_{0}|\sup_{0\leq\underline{u}\leq\delta}\left(\|\Omega\widehat{\chi}\|_{\mathbb{H}^{7}(\underline{u},u_{0})}+\|\omega\|_{\mathbb{H}^{5}(\underline{u},u_{0})}+\|L\phi\|_{\mathbb{H}^{5}(\underline{u},u_{0})}\right)\right\}<+\infty.\end{split} (3.3)

Then if the following smallness conditions hold:

Ω02(u0)δ|u1|12𝒲1,C2δ|u1|1𝒲𝒜1,formulae-sequencesuperscriptsubscriptΩ02subscript𝑢0𝛿superscriptsubscript𝑢11superscript2𝒲1superscript𝐶2𝛿superscriptsubscript𝑢11𝒲𝒜1\begin{split}\Omega_{0}^{2}(u_{0})\delta|u_{1}|^{-1}\mathscr{E}^{2}\mathscr{W}&\leq 1,\\ C^{2}\delta|u_{1}|^{-1}\mathscr{F}\mathscr{W}\mathcal{A}&\leq 1,\end{split} (3.4)

and the following auxiliary condition holds:

Ω02(u0)Ω02(u1)δ|u1|1𝒜1.superscriptsubscriptΩ02subscript𝑢0superscriptsubscriptΩ02subscript𝑢1𝛿superscriptsubscript𝑢11𝒜1\begin{split}\Omega_{0}^{2}(u_{0})\Omega_{0}^{-2}(u_{1})\delta|u_{1}|^{-1}\mathscr{F}\mathcal{A}&\leq 1.\end{split} (3.5)

Then the smooth solution of the Einstein-scalar field equations exists in the region 0u¯δ0¯𝑢𝛿0\leq\underline{u}\leq\delta, u0uu1subscript𝑢0𝑢subscript𝑢1u_{0}\leq u\leq u_{1}, and the following estimates hold:

𝒪,,𝒪~,~,𝒜.less-than-or-similar-to𝒪~𝒪~𝒜\mathcal{O},\mathcal{E},\widetilde{\mathcal{O}},\widetilde{\mathcal{E}},\mathcal{R}\lesssim\mathcal{A}.
Remark 3.2.

Here and in the rest of the paper, the notation ABless-than-or-similar-to𝐴𝐵A\lesssim B refers to AcB𝐴𝑐𝐵A\leq cB where c𝑐c is some universal constant.

Remark 3.3.

Using the argument in Chapter 2 of [9], under the assumptions of Theorem 3.1, we can obtain the initial bound

sup0u¯δsubscriptsupremum0¯𝑢𝛿\displaystyle\sup_{0\leq\underline{u}\leq\delta} (𝒪(u¯,u0;u0,u1)+(u¯,u0;u0,u1)+𝒪~(u¯,u0;u0,u1))𝒪¯𝑢subscript𝑢0subscript𝑢0subscript𝑢1¯𝑢subscript𝑢0subscript𝑢0subscript𝑢1~𝒪¯𝑢subscript𝑢0subscript𝑢0subscript𝑢1\displaystyle(\mathcal{O}(\underline{u},u_{0};u_{0},u_{1})+\mathcal{E}(\underline{u},u_{0};u_{0},u_{1})+\widetilde{\mathcal{O}}(\underline{u},u_{0};u_{0},u_{1}))
+~(u0;u0,u1)+(u0;u0,u1)𝒜(δ,u0,u1).less-than-or-similar-to~subscript𝑢0subscript𝑢0subscript𝑢1subscript𝑢0subscript𝑢0subscript𝑢1𝒜𝛿subscript𝑢0subscript𝑢1\displaystyle+\widetilde{\mathcal{E}}(u_{0};u_{0},u_{1})+\mathcal{R}(u_{0};u_{0},u_{1})\lesssim\mathcal{A}(\delta,u_{0},u_{1}).

We will omit the proof of this. This is however the bound we actually use in the proof of Theorem 3.1.

4. Proof of the existence theorem — Theorem 3.1

4.1. Bootstrap assumptions

As in [9], the main step to prove the existence is to establish a priori estimates. Once we have the a priori estimates, the construction of the solution in double null foliation then follows using a bootstrap argument. We should remark that although we are dealing with the Einstein-scalar field equations but not the vacuum Einstein equations, the argument in [9] can also be modified in the current case.

We begin the proof of the a priori estimates by assuming the smooth solution exists for 0u¯δ,u0uu1formulae-sequence0¯𝑢𝛿subscript𝑢0𝑢subscript𝑢10\leq\underline{u}\leq\delta,u_{0}\leq u\leq u_{1} and it holds for some CC0𝐶subscript𝐶0C\geq C_{0} the following bootstrap assumptions:

𝒪,,𝒪~,~,C14𝒜.less-than-or-similar-to𝒪~𝒪~superscript𝐶14𝒜\mathcal{O},\mathcal{E},\widetilde{\mathcal{O}},\widetilde{\mathcal{E}},\mathcal{R}\lesssim C^{\frac{1}{4}}\mathcal{A}. (4.1)

Then it suffices to prove that under this bootstrap assumptions (4.1), the smallness conditions (3.4) and the auxiliary condition (3.5), it holds

𝒪,,𝒪~,~,𝒜.less-than-or-similar-to𝒪~𝒪~𝒜\mathcal{O},\mathcal{E},\widetilde{\mathcal{O}},\widetilde{\mathcal{E}},\mathcal{R}\lesssim\mathcal{A}.

If C0subscript𝐶0C_{0} is sufficiently large, then this is an improvement over the bootstrap assumptions (4.1). We will prove this in the rest of this section.

4.2. Preliminary lemmas

We start by proving some geometric lemmas which will be frequently used in the course of the proof. First of all, we assume

|Ωtrχ2Ω2h|u||C34δ|u|22𝒜2,|Ωχ^|C12|u|1𝒜,14Ω0Ω4Ω0.formulae-sequenceless-than-or-similar-toΩtr𝜒2superscriptΩ2𝑢superscript𝐶34𝛿superscript𝑢2superscript2superscript𝒜2formulae-sequenceless-than-or-similar-toΩ^𝜒superscript𝐶12superscript𝑢1𝒜14subscriptΩ0Ω4subscriptΩ0\displaystyle\left|\Omega\mathrm{tr}\chi-\frac{2\Omega^{2}h}{|u|}\right|\lesssim C^{\frac{3}{4}}\delta|u|^{-2}\mathscr{F}^{2}\mathcal{A}^{2},\ |\Omega\widehat{\chi}|\lesssim C^{\frac{1}{2}}|u|^{-1}\mathscr{F}\mathcal{A},\ \frac{1}{4}\Omega_{0}\leq\Omega\leq 4\Omega_{0}. (4.2)

Let Λ(u¯;u)Λ¯𝑢𝑢\Lambda(\underline{u};u) and λ(u¯;u)𝜆¯𝑢𝑢\lambda(\underline{u};u) be the larger and smaller eigenvalues of the metric g/|Su¯,u\mbox{$g\mkern-9.0mu/$}|_{S_{\underline{u},u}} with respect to g/|S0,u\mbox{$g\mkern-9.0mu/$}|_{S_{0,u}}, which is the standard metric on the sphere with radius |u|𝑢|u|. Define

μ(u¯;u)=λ(u¯;u)Λ(u¯;u),ν(u¯;u)=Λ(u¯;u)λ(u¯;u).formulae-sequence𝜇¯𝑢𝑢𝜆¯𝑢𝑢Λ¯𝑢𝑢𝜈¯𝑢𝑢Λ¯𝑢𝑢𝜆¯𝑢𝑢\displaystyle\mu(\underline{u};u)=\sqrt{\lambda(\underline{u};u)\Lambda(\underline{u};u)},\ \nu(\underline{u};u)=\sqrt{\frac{\Lambda(\underline{u};u)}{\lambda(\underline{u};u)}}.

From the proof of Lemma 5.3 in [9], we have

μ(u¯;u)=exp(0u¯Ωtrχ(u¯,u)du¯),ν(u¯;u)exp(20u¯|Ωχ^(u¯,u)|du¯).formulae-sequence𝜇¯𝑢𝑢superscriptsubscript0¯𝑢Ωtr𝜒superscript¯𝑢𝑢differential-dsuperscript¯𝑢𝜈¯𝑢𝑢2superscriptsubscript0¯𝑢Ω^𝜒superscript¯𝑢𝑢differential-dsuperscript¯𝑢\displaystyle\mu(\underline{u};u)=\exp\left(\int_{0}^{\underline{u}}\Omega\mathrm{tr}\chi(\underline{u}^{\prime},u)\mathrm{d}\underline{u}^{\prime}\right),\ \nu(\underline{u};u)\leq\exp\left(2\int_{0}^{\underline{u}}|\Omega\widehat{\chi}(\underline{u}^{\prime},u)|\mathrm{d}\underline{u}^{\prime}\right).

From (4.2) and the smallness conditions (3.4), we have

|Ωtrχ||u|1+C1|u|1𝒜|u|1𝒜.less-than-or-similar-toΩtr𝜒superscript𝑢1superscript𝐶1superscript𝑢1𝒜less-than-or-similar-tosuperscript𝑢1𝒜\displaystyle|\Omega\mathrm{tr}\chi|\lesssim|u|^{-1}+C^{-1}|u|^{-1}\mathscr{F}\mathcal{A}\lesssim|u|^{-1}\mathscr{F}\mathcal{A}. (4.3)

Therefore, we have, using (4.2) and (3.4) again,

|μ(u¯;u)1|δ|u|1𝒜C1, 1ν(u¯;u)C12|u|1𝒜C1,formulae-sequenceless-than-or-similar-to𝜇¯𝑢𝑢1𝛿superscript𝑢1𝒜less-than-or-similar-tosuperscript𝐶11𝜈¯𝑢𝑢less-than-or-similar-tosuperscript𝐶12superscript𝑢1𝒜less-than-or-similar-tosuperscript𝐶1\displaystyle|\mu(\underline{u};u)-1|\lesssim\delta|u|^{-1}\mathscr{F}\mathcal{A}\lesssim C^{-1},\ 1\leq\nu(\underline{u};u)\lesssim C^{\frac{1}{2}}|u|^{-1}\mathscr{F}\mathcal{A}\lesssim C^{-1},

which implies

1cC01λ(u¯;u)Λ(u¯;u)1+cC01.1𝑐superscriptsubscript𝐶01𝜆¯𝑢𝑢Λ¯𝑢𝑢1𝑐superscriptsubscript𝐶01\displaystyle 1-cC_{0}^{-1}\leq\lambda(\underline{u};u)\leq\Lambda(\underline{u};u)\leq 1+cC_{0}^{-1}. (4.4)

By a similar argument of the proof of Lemma 5.4 in [9], we have

I(u¯,u)(1+cC01)I(0,u)12π(1+cC01)1𝐼¯𝑢𝑢1𝑐superscriptsubscript𝐶01𝐼0𝑢12𝜋1𝑐superscriptsubscript𝐶01less-than-or-similar-to1\displaystyle I(\underline{u},u)\leq(1+cC_{0}^{-1})I(0,u)\leq\frac{1}{2\pi}(1+cC_{0}^{-1})\lesssim 1 (4.5)

where I(u¯,u)𝐼¯𝑢𝑢I(\underline{u},u) is the isoperimetric constant of the sphere (Su¯,u,g/)(S_{\underline{u},u},\mbox{$g\mkern-9.0mu/$}) and C0subscript𝐶0C_{0} is sufficiently large. Also from (4.4), we have

1cC01Area(Su¯,u)Area(S0,u)1+cC01,1𝑐superscriptsubscript𝐶01Areasubscript𝑆¯𝑢𝑢Areasubscript𝑆0𝑢1𝑐superscriptsubscript𝐶01\displaystyle 1-cC_{0}^{-1}\leq\frac{\mathrm{Area}(S_{\underline{u},u})}{\mathrm{Area}(S_{0,u})}\leq 1+cC_{0}^{-1},

which implies if C0subscript𝐶0C_{0} is sufficiently large,

12|u|214πArea(Su¯,u)2|u|2.12superscript𝑢214𝜋Areasubscript𝑆¯𝑢𝑢2superscript𝑢2\displaystyle\frac{1}{2}|u|^{2}\leq\frac{1}{4\pi}\mathrm{Area}(S_{\underline{u},u})\leq 2|u|^{2}. (4.6)

Now by Lemma 5.1 in [9], the Sobolev inequalities, we have the following form of the Sobolev inequalities:

Lemma 4.1.

Given a tangential tensorfield θ𝜃\theta, we have for q(2,+)𝑞2q\in(2,+\infty),

θ𝕃q(u¯,u)subscriptnorm𝜃superscript𝕃𝑞¯𝑢𝑢\displaystyle\|\theta\|_{\mathbb{L}^{q}(\underline{u},u)} qθ1(u¯,u),subscriptless-than-or-similar-to𝑞absentsubscriptnorm𝜃superscript1¯𝑢𝑢\displaystyle\lesssim_{q}\|\theta\|_{\mathbb{H}^{1}(\underline{u},u)},
θ𝕃(u¯,u)subscriptnorm𝜃superscript𝕃¯𝑢𝑢\displaystyle\|\theta\|_{\mathbb{L}^{\infty}(\underline{u},u)} (|u|/ )θ𝕃4(u¯,u)+θ𝕃4(u¯,u)θ2(u¯,u).less-than-or-similar-toabsentsubscriptnorm𝑢/ 𝜃superscript𝕃4¯𝑢𝑢subscriptnorm𝜃superscript𝕃4¯𝑢𝑢less-than-or-similar-tosubscriptnorm𝜃superscript2¯𝑢𝑢\displaystyle\lesssim\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\theta\|_{\mathbb{L}^{4}(\underline{u},u)}+\|\theta\|_{\mathbb{L}^{4}(\underline{u},u)}\lesssim\|\theta\|_{\mathbb{H}^{2}(\underline{u},u)}.

Here AqBsubscriptless-than-or-similar-to𝑞𝐴𝐵A\lesssim_{q}B means AcqB𝐴subscript𝑐𝑞𝐵A\leq c_{q}B where cqsubscript𝑐𝑞c_{q} is a constant depending only on q𝑞q.

By the classical Hölder inequality and the Sobolev inequalities, we have

Lemma 4.2.

Given tangential tensorfields θ1,,θnsubscript𝜃1subscript𝜃𝑛\theta_{1},\cdots,\theta_{n}, for i2𝑖2i\geq 2, we have

θ1θni(u¯,u)nθ1i(u¯,u)θni(u¯,u).subscriptless-than-or-similar-to𝑛subscriptnormsubscript𝜃1subscript𝜃𝑛superscript𝑖¯𝑢𝑢subscriptnormsubscript𝜃1superscript𝑖¯𝑢𝑢subscriptnormsubscript𝜃𝑛superscript𝑖¯𝑢𝑢\displaystyle\|\theta_{1}\cdots\theta_{n}\|_{\mathbb{H}^{i}(\underline{u},u)}\lesssim_{n}\|\theta_{1}\|_{\mathbb{H}^{i}(\underline{u},u)}\cdots\|\theta_{n}\|_{\mathbb{H}^{i}(\underline{u},u)}.

We then introduce the following Gronwall type estimates which are need to estimate using the transport equations:

Lemma 4.3.

For an s𝑠s-covariant tengential tensorfield θ𝜃\theta, and any real number ν𝜈\nu, we have

θ𝕃2(u¯,u)less-than-or-similar-tosubscriptnorm𝜃superscript𝕃2¯𝑢𝑢absent\displaystyle\|\theta\|_{\mathbb{L}^{2}(\underline{u},u)}\lesssim (θ|𝕃2(0,u)+δDθ𝕃u¯1𝕃2(u))s.{}_{s}\left(\|\theta|_{\mathbb{L}^{2}(0,u)}+\delta\|D\theta\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{L}^{2}(u)}\right).
|u|s+νθ𝕃q(u¯,u)less-than-or-similar-tosubscriptnormsuperscript𝑢𝑠𝜈𝜃superscript𝕃𝑞¯𝑢𝑢absent\displaystyle\||u|^{s+\nu}\theta\|_{\mathbb{L}^{q}(\underline{u},u)}\lesssim (|u|s+νθ𝕃q(u¯,u0)+|u|s+ν+1(D¯θ+ν2Ωtrχ¯θ)𝕃[u0,u]1𝕃2(u¯))s,ν,{}_{s,\nu}\left(\||u|^{s+\nu}\theta\|_{\mathbb{L}^{q}(\underline{u},u_{0})}+\||u|^{s+\nu+1}(\underline{D}\theta+\frac{\nu}{2}\Omega\mathrm{tr}\underline{\chi}\theta)\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{L}^{2}(\underline{u})}\right),
Proof.

The first inequality is similar to those in Lemma 4.4 and 4.6 in [9]. Using the argument in [9], the first inequality holds if for any u¯,u¯[0,δ]superscript¯𝑢¯𝑢0𝛿\underline{u}^{\prime},\underline{u}\in[0,\delta],

u¯u¯Ωtrχdu¯′′,u¯u¯|Ωχ^|du¯′′1.less-than-or-similar-tosuperscriptsubscriptsuperscript¯𝑢¯𝑢Ωtr𝜒differential-dsuperscript¯𝑢′′superscriptsubscriptsuperscript¯𝑢¯𝑢Ω^𝜒differential-dsuperscript¯𝑢′′1\displaystyle\int_{\underline{u}^{\prime}}^{\underline{u}}\Omega\mathrm{tr}\chi\mathrm{d}\underline{u}^{\prime\prime},\int_{\underline{u}^{\prime}}^{\underline{u}}|\Omega\widehat{\chi}|\mathrm{d}\underline{u}^{\prime\prime}\lesssim 1.

This is true because both of them are bounded by,

0δC12|u|1𝒜du¯C12δ|u|1𝒜C1,less-than-or-similar-tosuperscriptsubscript0𝛿superscript𝐶12superscript𝑢1𝒜differential-dsuperscript¯𝑢superscript𝐶12𝛿superscript𝑢1𝒜less-than-or-similar-tosuperscript𝐶1\displaystyle\int_{0}^{\delta}C^{\frac{1}{2}}|u|^{-1}\mathscr{F}\mathcal{A}\mathrm{d}\underline{u}^{\prime}\lesssim C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\lesssim C^{-1},

where we use (4.2) and the smallness conditions (3.4).

The second inequality is similar to those in Lemma 4.5 and 4.7 in [9]. It holds if for any u,u[u0,u1]superscript𝑢𝑢subscript𝑢0subscript𝑢1u^{\prime},u\in[u_{0},u_{1}],

uu(Ωtrχ¯+2|u|)du′′,uu|Ωχ¯^|du′′1.less-than-or-similar-tosuperscriptsubscriptsuperscript𝑢𝑢Ωtr¯𝜒2𝑢differential-dsuperscript𝑢′′superscriptsubscriptsuperscript𝑢𝑢Ω^¯𝜒differential-dsuperscript𝑢′′1\displaystyle\int_{u^{\prime}}^{u}\left(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\right)\mathrm{d}u^{\prime\prime},\int_{u^{\prime}}^{u}|\Omega\widehat{\underline{\chi}}|\mathrm{d}u^{\prime\prime}\lesssim 1.

This is true because, from the Sobolev ienqualities, Lemma 4.1,

|Ωtrχ¯+2|u||,|Ωχ¯^|C14δ|u|2𝒜,less-than-or-similar-toΩtr¯𝜒2𝑢Ω^¯𝜒superscript𝐶14𝛿superscript𝑢2𝒜\displaystyle\left|\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\right|,|\Omega\widehat{\underline{\chi}}|\lesssim C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathcal{A},

and then both of the above integrals are bounded by

uuC14δ|u|2𝒜du′′C14δ|u|1𝒜C1.less-than-or-similar-tosuperscriptsubscriptsuperscript𝑢𝑢superscript𝐶14𝛿superscript𝑢2𝒜differential-dsuperscript𝑢′′superscript𝐶14𝛿superscript𝑢1𝒜less-than-or-similar-tosuperscript𝐶1\displaystyle\int_{u^{\prime}}^{u}C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathcal{A}\mathrm{d}u^{\prime\prime}\lesssim C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\lesssim C^{-1}.

 

In addition, we have the Gronwall type estimates for the augular derivatives.

Lemma 4.4.

For an s𝑠s-covariant tengential tensorfield θ𝜃\theta, and any real number ν𝜈\nu, we have, for n4𝑛4n\leq 4 (if s=0𝑠0s=0, then n5𝑛5n\leq 5)

θn(u¯,u)less-than-or-similar-tosubscriptnorm𝜃superscript𝑛¯𝑢𝑢absent\displaystyle\|\theta\|_{\mathbb{H}^{n}(\underline{u},u)}\lesssim sθn(0,u)+δDθ𝕃u¯1n(u).{}_{s}\|\theta\|_{\mathbb{H}^{n}(0,u)}+\delta\|D\theta\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{n}(u)}.
|u|s+νθn(u¯,u)less-than-or-similar-tosubscriptnormsuperscript𝑢𝑠𝜈𝜃superscript𝑛¯𝑢𝑢absent\displaystyle\||u|^{s+\nu}\theta\|_{\mathbb{H}^{n}(\underline{u},u)}\lesssim (|u|s+νθn(u¯,u0)+|u|s+ν+1(D¯θ+ν2Ωtrχ¯θ)𝕃[u0,u]1n(u¯))s,ν,{}_{s,\nu}\left(\||u|^{s+\nu}\theta\|_{\mathbb{H}^{n}(\underline{u},u_{0})}+\||u|^{s+\nu+1}(\underline{D}\theta+\frac{\nu}{2}\Omega\mathrm{tr}\underline{\chi}\theta)\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{n}(\underline{u})}\right),
Proof.

For the first inequality, we apply the above Gronwall estimate, Lemma 4.3 to the following equation:

D/ iθ=/ iDθ+[D,/ i]θ=/ iDθ+j=1i/ j(Ωχ)/ ijθ.𝐷superscript/ 𝑖𝜃superscript/ 𝑖𝐷𝜃𝐷superscript/ 𝑖𝜃superscript/ 𝑖𝐷𝜃superscriptsubscript𝑗1𝑖superscript/ 𝑗Ω𝜒superscript/ 𝑖𝑗𝜃\displaystyle D\mbox{$\nabla\mkern-13.0mu/$ }^{i}\theta=\mbox{$\nabla\mkern-13.0mu/$ }^{i}D\theta+[D,\mbox{$\nabla\mkern-13.0mu/$ }^{i}]\theta=\mbox{$\nabla\mkern-13.0mu/$ }^{i}D\theta+\sum_{j=1}^{i}\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\Omega\chi)\mbox{$\nabla\mkern-13.0mu/$ }^{i-j}\theta.

By Hölder inequality and Sobolev inequalies,

θn(u¯,u)θn(0,u)+δDθ𝕃u¯1n(u)+δΩχ𝕃u¯n(u¯)θ𝕃u¯n(u),less-than-or-similar-tosubscriptnorm𝜃superscript𝑛¯𝑢𝑢subscriptnorm𝜃superscript𝑛0𝑢𝛿subscriptnorm𝐷𝜃superscriptsubscript𝕃¯𝑢1superscript𝑛𝑢𝛿subscriptnormΩ𝜒subscriptsuperscript𝕃¯𝑢superscript𝑛¯𝑢subscriptnorm𝜃subscriptsuperscript𝕃¯𝑢superscript𝑛𝑢\displaystyle\|\theta\|_{\mathbb{H}^{n}(\underline{u},u)}\lesssim\|\theta\|_{\mathbb{H}^{n}(0,u)}+\delta\|D\theta\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{n}(u)}+\delta\|\Omega\chi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{n}(\underline{u})}\|\theta\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{n}(u)},

where we have used the following Hölder type inequality:

𝕃u¯1𝕃u¯2𝕃u¯.\displaystyle\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}}\lesssim\|\cdot\|_{\mathbb{L}^{2}_{\underline{u}}}\lesssim\|\cdot\|_{\mathbb{L}^{\infty}_{\underline{u}}}. (4.7)

By (4.3), the bootstrap assumptions (4.1) on Ωχ^Ω^𝜒\Omega\widehat{\chi} and the smallness conditions (3.4), we have, for n4𝑛4n\leq 4,

θn(u¯,u)less-than-or-similar-tosubscriptnorm𝜃superscript𝑛¯𝑢𝑢absent\displaystyle\|\theta\|_{\mathbb{H}^{n}(\underline{u},u)}\lesssim θn(0,u)+δDθ𝕃u¯1n(u)+C14δ|u|1𝒜θ𝕃u¯n(u)subscriptnorm𝜃superscript𝑛0𝑢𝛿subscriptnorm𝐷𝜃superscriptsubscript𝕃¯𝑢1superscript𝑛𝑢superscript𝐶14𝛿superscript𝑢1𝒜subscriptnorm𝜃subscriptsuperscript𝕃¯𝑢superscript𝑛𝑢\displaystyle\|\theta\|_{\mathbb{H}^{n}(0,u)}+\delta\|D\theta\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{n}(u)}+C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\|\theta\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{n}(u)}
less-than-or-similar-to\displaystyle\lesssim θn(0,u)+δDθ𝕃u¯1n(u)+C1θ𝕃u¯n(u),subscriptnorm𝜃superscript𝑛0𝑢𝛿subscriptnorm𝐷𝜃superscriptsubscript𝕃¯𝑢1superscript𝑛𝑢superscript𝐶1subscriptnorm𝜃subscriptsuperscript𝕃¯𝑢superscript𝑛𝑢\displaystyle\|\theta\|_{\mathbb{H}^{n}(0,u)}+\delta\|D\theta\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{n}(u)}+C^{-1}\|\theta\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{n}(u)},

Then if C0subscript𝐶0C_{0} is sufficiently large, the last term on the right hand side can be absorbed by the left hand side after taking supremum on u¯¯𝑢\underline{u}. We then obtain the desired inequality. If s=0𝑠0s=0, that is, when θ𝜃\theta is a function, then D/ θ=/ Dθ𝐷/ 𝜃/ 𝐷𝜃D\mbox{$\nabla\mkern-13.0mu/$ }\theta=\mbox{$\nabla\mkern-13.0mu/$ }D\theta. Therefore [D,/ 5]θ𝐷superscript/ 5𝜃[D,\mbox{$\nabla\mkern-13.0mu/$ }^{5}]\theta contains only four derivatives of ΩχΩ𝜒\Omega\chi.

The second inequality can be obtained similarly.  

We then introduce an estimate for ΩΩ\Omega and its derivatives:

Lemma 4.5.
12Ω0Ω2Ω0,12subscriptΩ0Ω2subscriptΩ0\frac{1}{2}\Omega_{0}\leq\Omega\leq 2\Omega_{0}, (4.8)

and

Ωs4(u¯,u)sΩ0s.subscriptless-than-or-similar-to𝑠subscriptnormsuperscriptΩ𝑠superscript4¯𝑢𝑢superscriptsubscriptΩ0𝑠\|\Omega^{s}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim_{s}\Omega_{0}^{s}.

Moreover, we have

ΩsΩ0s4(u¯,u)sC14Ω0sδ|u|1𝒲12𝒜.subscriptless-than-or-similar-to𝑠subscriptnormsuperscriptΩ𝑠superscriptsubscriptΩ0𝑠superscript4¯𝑢𝑢superscript𝐶14superscriptsubscriptΩ0𝑠𝛿superscript𝑢1superscript𝒲12𝒜\|\Omega^{s}-\Omega_{0}^{s}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim_{s}C^{\frac{1}{4}}\Omega_{0}^{s}\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (4.9)
Proof.

Since DlogΩ=ω𝐷Ω𝜔D\log{\Omega}=\omega, we then write D(logΩlogΩ0)=ω𝐷ΩsubscriptΩ0𝜔D(\log\Omega-\log\Omega_{0})=\omega. Applying Lemma 4.3,

logΩlogΩ04(u¯,u)δω𝕃u¯14(u)C14δ|u|1𝒲12𝒜C1.less-than-or-similar-tosubscriptnormΩsubscriptΩ0superscript4¯𝑢𝑢𝛿subscriptnorm𝜔superscriptsubscript𝕃¯𝑢1superscript4𝑢less-than-or-similar-tosuperscript𝐶14𝛿superscript𝑢1superscript𝒲12𝒜less-than-or-similar-tosuperscript𝐶1\|\log{\Omega}-\log{\Omega_{0}}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\delta\|\omega\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{4}(u)}\lesssim C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-1}.

In particular, if C0subscript𝐶0C_{0} is sufficiently large, we have, by the Sobolev inequalities,

|logΩΩ0|log2ΩsubscriptΩ02\left|\log\frac{\Omega}{\Omega_{0}}\right|\leq\log 2

which implies the first desired inequality. For the second, if s0𝑠0s\neq 0, then

Ωs4(u¯,u)sΩs𝕃(u¯,u)(1+logΩlogΩ04(u¯,u)4)Ω0s.subscriptless-than-or-similar-to𝑠subscriptnormsuperscriptΩ𝑠superscript4¯𝑢𝑢subscriptnormsuperscriptΩ𝑠superscript𝕃¯𝑢𝑢1superscriptsubscriptnormΩsubscriptΩ0superscript4¯𝑢𝑢4less-than-or-similar-tosuperscriptsubscriptΩ0𝑠\displaystyle\|\Omega^{s}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim_{s}\|\Omega^{s}\|_{\mathbb{L}^{\infty}(\underline{u},u)}(1+\|\log{\Omega}-\log\Omega_{0}\|_{\mathbb{H}^{4}(\underline{u},u)}^{4})\lesssim\Omega_{0}^{s}.

The last estimate (4.9) follows similarly from the equation DΩs=sΩsω𝐷superscriptΩ𝑠𝑠superscriptΩ𝑠𝜔D\Omega^{s}=s\Omega^{s}\omega and the bootstrap assumptions (4.1).  

This lemma implies that for a tangential tensorfield θ𝜃\theta,

Ωsθn(u¯,u)Ω0sθn(u¯,u)less-than-or-similar-tosubscriptnormsuperscriptΩ𝑠𝜃superscript𝑛¯𝑢𝑢superscriptsubscriptΩ0𝑠subscriptnorm𝜃superscript𝑛¯𝑢𝑢\displaystyle\|\Omega^{s}\theta\|_{\mathbb{H}^{n}(\underline{u},u)}\lesssim\Omega_{0}^{s}\|\theta\|_{\mathbb{H}^{n}(\underline{u},u)} (4.10)

for n4𝑛4n\leq 4. This inequality says that we do not need to worry about the appearance of ΩΩ\Omega if we take less than four derivatives.

This lemma also improves the bound for ΩΩ\Omega in (4.2). The bounds for ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi and Ωχ^Ω^𝜒\Omega\widehat{\chi} can also be improved by using the bootstrap assumptions (4.1) on 𝒪𝒪\mathcal{O} and the Sobolev inequalities, Lemma 4.1. The constants C34,C12superscript𝐶34superscript𝐶12C^{\frac{3}{4}},C^{\frac{1}{2}} in (4.2) can be improved to C12,C14superscript𝐶12superscript𝐶14C^{\frac{1}{2}},C^{\frac{1}{4}} respectively if C0subscript𝐶0C_{0} is sufficiently large. This implies that all conclusions in this subsection hold without assuming (4.2).

Finally, we introduce the elliptic estimates.

Lemma 4.6.

Suppose that

K2(u¯,u)|u|2.less-than-or-similar-tosubscriptnorm𝐾superscript2¯𝑢𝑢superscript𝑢2\displaystyle\|K\|_{\mathbb{H}^{2}(\underline{u},u)}\lesssim|u|^{-2}.

Now assume that θ𝜃\theta is a tangential symmetric trace-free (0,2)02(0,2) type tensorfield with

div/ θ=f,div/ 𝜃𝑓\mbox{$\mathrm{div}\mkern-13.0mu/$ }\theta=f,

where f𝑓f is a tangential one-form. Then

θ5(u¯,u)|u|f4(u¯,u)+(1+|u|2K3(u¯,u))θ2(u¯,u).less-than-or-similar-tosubscriptnorm𝜃superscript5¯𝑢𝑢subscriptnorm𝑢𝑓superscript4¯𝑢𝑢1subscriptnormsuperscript𝑢2𝐾superscript3¯𝑢𝑢subscriptnorm𝜃superscript2¯𝑢𝑢\displaystyle\|\theta\|_{\mathbb{H}^{5}(\underline{u},u)}\lesssim\||u|f\|_{\mathbb{H}^{4}(\underline{u},u)}+(1+\||u|^{2}K\|_{\mathbb{H}^{3}(\underline{u},u)})\|\theta\|_{\mathbb{H}^{2}(\underline{u},u)}.

Assume that ξ𝜉\xi is a tangential (0,1)01(0,1) type tensorfield with

div/ ξ=f,curl/ ξ=gformulae-sequencediv/ 𝜉𝑓curl/ 𝜉𝑔\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }\xi=f,\mbox{$\mathrm{curl}\mkern-13.0mu/$ }\xi=g

where f,g𝑓𝑔f,g are functions. Then

ξ5(u¯,u)|u|(f,g)4(u¯,u)+(1+|u|2K3(u¯,u))ξ4(u¯,u).less-than-or-similar-tosubscriptnorm𝜉superscript5¯𝑢𝑢subscriptnorm𝑢𝑓𝑔superscript4¯𝑢𝑢1subscriptnormsuperscript𝑢2𝐾superscript3¯𝑢𝑢subscriptnorm𝜉superscript4¯𝑢𝑢\displaystyle\|\xi\|_{\mathbb{H}^{5}(\underline{u},u)}\lesssim\||u|(f,g)\|_{\mathbb{H}^{4}(\underline{u},u)}+(1+\||u|^{2}K\|_{\mathbb{H}^{3}(\underline{u},u)})\|\xi\|_{\mathbb{H}^{4}(\underline{u},u)}.

Assume that ϕitalic-ϕ\phi is a function with

Δ/ ϕ=f.Δ/ italic-ϕ𝑓\mbox{$\Delta\mkern-13.0mu/$ }\phi=f.

We have

(|u|/ )ϕ4(u¯,u)|u|2f3(u¯,u).less-than-or-similar-tosubscriptnorm𝑢/ italic-ϕsuperscript4¯𝑢𝑢subscriptnormsuperscript𝑢2𝑓superscript3¯𝑢𝑢\displaystyle\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\phi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\||u|^{2}f\|_{\mathbb{H}^{3}(\underline{u},u)}.
Proof.

The first and second conclusions are obtained by repeatedly use the argument of Section 7.3 in [9]. For the last conclusion, we apply the second conclusion for the one form / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi for up to the fourth order derivatives. Therefore K3(u¯,u)subscriptnorm𝐾superscript3¯𝑢𝑢\|K\|_{\mathbb{H}^{3}(\underline{u},u)} does not come in and

(|u|/ )ϕ4(u¯,u)|u|2f3(u¯,u)+(|u|/ )ϕ𝕃2(u¯,u)less-than-or-similar-tosubscriptnorm𝑢/ italic-ϕsuperscript4¯𝑢𝑢subscriptnormsuperscript𝑢2𝑓superscript3¯𝑢𝑢subscriptnorm𝑢/ italic-ϕsuperscript𝕃2¯𝑢𝑢\displaystyle\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\phi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\||u|^{2}f\|_{\mathbb{H}^{3}(\underline{u},u)}+\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\phi\|_{\mathbb{L}^{2}(\underline{u},u)}

Integration by parts gives

Su¯,u|/ ϕ|2dμg/Su¯,u|f|2dμg/.\int_{S_{\underline{u},u}}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim\int_{S_{\underline{u},u}}|f|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}.

These two inequalities give the desired conclusion.  

We end this section by the following estimates for ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi, Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} and L¯ϕ¯𝐿italic-ϕ\underline{L}\phi. The zeroth order bounds of these quantities are worse than their derivatives.

Lemma 4.7.

Under the assumptions of the Theorem 3.1 and the bootstrap assumptions (4.1), we have

|u|Ωtrχ4(u¯,u)𝒜,|u|Ωtrχ¯4(u¯,u)1,|u|L¯ϕ𝕃[u0,u]24(u¯)𝒲12.formulae-sequenceless-than-or-similar-to𝑢subscriptnormΩtr𝜒superscript4¯𝑢𝑢𝒜formulae-sequenceless-than-or-similar-to𝑢subscriptnormΩtr¯𝜒superscript4¯𝑢𝑢1less-than-or-similar-tosubscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢superscript𝒲12\displaystyle|u|\|\Omega\mathrm{tr}\chi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\mathscr{F}\mathcal{A},\ |u|\|\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim 1,\ \||u|\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\mathscr{W}^{\frac{1}{2}}.
Proof.

From the bootstrap assumptions (4.1) and the smallness conditions (3.4):

|u|Ωtrχ4(u¯,u)less-than-or-similar-to𝑢subscriptnormΩtr𝜒superscript4¯𝑢𝑢absent\displaystyle|u|\|\Omega\mathrm{tr}\chi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim |u|Ω02trχ4(u¯,u)1+C12δ|u|12𝒜2(1+C1)𝒜𝒜,less-than-or-similar-to𝑢subscriptnormsuperscriptsubscriptΩ02trsuperscript𝜒superscript4¯𝑢𝑢1superscript𝐶12𝛿superscript𝑢1superscript2superscript𝒜2less-than-or-similar-to1superscript𝐶1𝒜less-than-or-similar-to𝒜\displaystyle|u|\|\Omega_{0}^{2}\mathrm{tr}\chi^{\prime}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim 1+C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^{2}\mathcal{A}^{2}\lesssim(1+C^{-1})\mathscr{F}\mathcal{A}\lesssim\mathscr{F}\mathcal{A}, (4.11)
|u|Ωtrχ¯4(u¯,u)less-than-or-similar-to𝑢subscriptnormΩtr¯𝜒superscript4¯𝑢𝑢absent\displaystyle|u|\|\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim 1+C14δ|u|1𝒜(1+C1)1,less-than-or-similar-to1superscript𝐶14𝛿superscript𝑢1𝒜1superscript𝐶1less-than-or-similar-to1\displaystyle 1+C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\lesssim(1+C^{-1})\lesssim 1, (4.12)
|u|L¯ϕ𝕃[u0,u]24(u¯)(u0u|ψ|2|u|du)12+|u|14|u|34(|u|L¯ϕψ)𝕃[u0,u]24(u¯)|logΩ0(u1)Ω0(u0)|12+C14δ|u|1𝒜𝒲12.less-than-or-similar-tosubscriptdelimited-∥∥𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢12superscript𝑢14subscriptdelimited-∥∥superscript𝑢34𝑢¯𝐿italic-ϕ𝜓subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-tosuperscriptsubscriptΩ0subscript𝑢1subscriptΩ0subscript𝑢012superscript𝐶14𝛿superscript𝑢1𝒜less-than-or-similar-tosuperscript𝒲12\displaystyle\begin{split}\||u|\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim&\left(\int_{u_{0}}^{u}\frac{|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}+|u|^{-\frac{1}{4}}\||u|^{\frac{3}{4}}(|u|\underline{L}\phi-\psi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\\ \lesssim&\left|\log\frac{\Omega_{0}(u_{1})}{\Omega_{0}(u_{0})}\right|^{\frac{1}{2}}+C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\lesssim\mathscr{W}^{\frac{1}{2}}.\end{split} (4.13)

 

We have finished stating and proving the preliminary lemmas and begin to estimate the scalar field, the connection coefficients and the curvature components. From now on, we will frequently use the bootstrap assumptions (4.1) and the smallness conditions (3.4) and we may not point this out in the text.

4.3. Estimates for \mathcal{E}

We first estimate the lower order derivatives of the derivative of the wave function ϕitalic-ϕ\phi.

Proposition 4.1.

Under the assumptions of Theorem 3.1 and the bootstrap assumptions (4.1), we have

𝒜+~.less-than-or-similar-to𝒜~\mathcal{E}\lesssim\mathcal{A}+\widetilde{\mathcal{E}}.
Proof.

Estimate for Lϕ𝐿italic-ϕL\phi: The estimate for Lϕ𝐿italic-ϕL\phi makes use of the following equation, which is rewritten in terms of Lϕφ/|u|𝐿italic-ϕ𝜑𝑢L\phi-\varphi/|u|:

D¯(Lϕφ/|u|)+12Ωtrχ¯(Lϕφ/|u|)=Ω2Δ/ ϕ+2Ω2(η,/ ϕ)12(Ωtrχ¯+2|u|)φ|u|12Ωtrχ(L¯ϕψ/|u|)12(Ωtrχ2Ω2h|u|)ψ|u|(Ω2Ω02)hψ|u|2,¯𝐷𝐿italic-ϕ𝜑𝑢12Ωtr¯𝜒𝐿italic-ϕ𝜑𝑢superscriptΩ2Δ/ italic-ϕ2superscriptΩ2𝜂/ italic-ϕ12Ωtr¯𝜒2𝑢𝜑𝑢12Ωtr𝜒¯𝐿italic-ϕ𝜓𝑢12Ωtr𝜒2superscriptΩ2𝑢𝜓𝑢superscriptΩ2superscriptsubscriptΩ02𝜓superscript𝑢2\begin{split}&\underline{D}(L\phi-\varphi/|u|)+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}(L\phi-\varphi/|u|)=\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}(\eta,\mbox{$\nabla\mkern-13.0mu/$ }\phi)\\ &-\frac{1}{2}(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})\frac{\varphi}{|u|}-\frac{1}{2}\Omega\mathrm{tr}\chi(\underline{L}\phi-\psi/|u|)-\frac{1}{2}(\Omega\mathrm{tr}\chi-\frac{2\Omega^{2}h}{|u|})\frac{\psi}{|u|}-(\Omega^{2}-\Omega_{0}^{2})\frac{h\psi}{|u|^{2}},\\ \end{split} (4.14)

To estimate the 4(u¯,u)superscript4¯𝑢𝑢\mathbb{H}^{4}(\underline{u},u) norm of Lϕφ/|u|𝐿italic-ϕ𝜑𝑢L\phi-\varphi/|u|, we need to estimate the right hand side in |u|2𝕃[u0,u]14(u¯)\||u|^{2}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}. We estimate them term by term.

The \engordnumber1 term on the right hand side is estimated by

Ω0(u0)|u|12Ω0|u|32(|u|/ )/ ϕ𝕃[u0,u]24(u¯)~,less-than-or-similar-toabsentsubscriptΩ0subscript𝑢0superscript𝑢12subscriptnormsubscriptΩ0superscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-to~\displaystyle\lesssim\Omega_{0}(u_{0})|u|^{-\frac{1}{2}}\|\Omega_{0}|u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\mathscr{F}\widetilde{\mathcal{E}},

where we have used a frequently used the following Hölder type inequality for s>0𝑠0s>0:

|u|s𝕃[u0,u]1|u|s𝕃[u0,u]2𝕃[u0,u]2|u|s𝕃[u0,u]2.\displaystyle\||u|^{-s}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}}\lesssim\||u|^{-s}\|_{\mathbb{L}^{2}_{[u_{0},u]}}\|\cdot\|_{\mathbb{L}^{2}_{[u_{0},u]}}\lesssim|u|^{-s}\|\cdot\|_{\mathbb{L}^{2}_{[u_{0},u]}}. (4.15)

The \engordnumber2 term on the right hand side is estimated by

Ω02(u0)|u|2|u|2η𝕃[u0,u]4(u¯)|u|2/ ϕ𝕃[u0,u]4(u¯)Ω02(u0)C12δ2|u|222𝒜2C1𝒜.less-than-or-similar-toabsentsuperscriptsubscriptΩ02subscript𝑢0superscript𝑢2subscriptnormsuperscript𝑢2𝜂subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-tosuperscriptsubscriptΩ02subscript𝑢0superscript𝐶12superscript𝛿2superscript𝑢2superscript2superscript2superscript𝒜2less-than-or-similar-tosuperscript𝐶1𝒜\displaystyle\lesssim\Omega_{0}^{2}(u_{0})|u|^{-2}\||u|^{2}\eta\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\Omega_{0}^{2}(u_{0})C^{\frac{1}{2}}\delta^{2}|u|^{-2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\lesssim C^{-1}\mathscr{F}\mathcal{A}.

The \engordnumber3 term on the right hand side is estimated by

|u|1(supu0uu1|φ(u)|)|u|2(Ωtrχ¯+2|u|)𝕃[u0,u]4(u¯)C14δ|u|12𝒜C1𝒜.less-than-or-similar-toabsentsuperscript𝑢1subscriptsupremumsubscript𝑢0superscript𝑢subscript𝑢1𝜑superscript𝑢subscriptnormsuperscript𝑢2Ωtr¯𝜒2𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-tosuperscript𝐶14𝛿superscript𝑢1superscript2𝒜less-than-or-similar-tosuperscript𝐶1𝒜\displaystyle\lesssim|u|^{-1}\left(\sup_{u_{0}\leq u^{\prime}\leq u_{1}}|\varphi(u^{\prime})|\right)\left\||u|^{2}\left(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\right)\right\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}^{2}\mathcal{A}\lesssim C^{-1}\mathscr{F}\mathcal{A}.

The \engordnumber4 term on the right hand side is estimated by (using (4.11))

less-than-or-similar-to\displaystyle\lesssim |u|1|u|Ωtrχ𝕃[u0,u]4(u¯)|u|14|u|34(|u|L¯ϕψ)𝕃[u0,u]24(u¯)superscript𝑢1subscriptnorm𝑢Ωtr𝜒subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢superscript𝑢14subscriptnormsuperscript𝑢34𝑢¯𝐿italic-ϕ𝜓subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢\displaystyle|u|^{-1}\||u|\Omega\mathrm{tr}\chi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}|u|^{\frac{1}{4}}\||u|^{\frac{3}{4}}(|u|\underline{L}\phi-\psi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim C14δ|u|1𝒜𝒜C1𝒜.less-than-or-similar-tosuperscript𝐶14𝛿superscript𝑢1𝒜𝒜superscript𝐶1𝒜\displaystyle C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\cdot\mathscr{F}\mathcal{A}\lesssim C^{-1}\mathscr{F}\mathcal{A}.

The \engordnumber5 term on the right hand side is estimated by

less-than-or-similar-to\displaystyle\lesssim |u|1|u|2Ω02(trχ2h|u|2)𝕃[u0,u]4(u¯)(u0u|ψ|2|u|du)12superscript𝑢1subscriptnormsuperscript𝑢2superscriptsubscriptΩ02trsuperscript𝜒2superscript𝑢2subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢12\displaystyle|u|^{-1}\left\||u|^{2}\Omega_{0}^{2}\left(\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|^{2}}\right)\right\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\left(\int_{u_{0}}^{u}\frac{|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}
less-than-or-similar-to\displaystyle\lesssim C12δ|u|12|logΩ(u1)Ω(u0)|12𝒜2C1𝒜.less-than-or-similar-tosuperscript𝐶12𝛿superscript𝑢1superscript2superscriptΩsubscript𝑢1Ωsubscript𝑢012superscript𝒜2superscript𝐶1𝒜\displaystyle C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^{2}\left|\log\frac{\Omega(u_{1})}{\Omega(u_{0})}\right|^{\frac{1}{2}}\mathcal{A}^{2}\lesssim C^{-1}\mathscr{F}\mathcal{A}.

At last, the \engordnumber6 term on the right hand side is estimated by, using (4.9),

less-than-or-similar-to\displaystyle\lesssim |u|1|u|(Ω2Ω02)𝕃[u0,u]4(u¯)(u0u|ψ|2|u|du)12superscript𝑢1subscriptnorm𝑢superscriptΩ2superscriptsubscriptΩ02subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢12\displaystyle|u|^{-1}\left\||u|(\Omega^{2}-\Omega_{0}^{2})\right\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\left(\int_{u_{0}}^{u}\frac{|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}
less-than-or-similar-to\displaystyle\lesssim C14δ|u|1|logΩ(u1)Ω(u0)|12𝒲12𝒜C1𝒜.less-than-or-similar-tosuperscript𝐶14𝛿superscript𝑢1superscriptΩsubscript𝑢1Ωsubscript𝑢012superscript𝒲12𝒜superscript𝐶1𝒜\displaystyle C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\left|\log\frac{\Omega(u_{1})}{\Omega(u_{0})}\right|^{\frac{1}{2}}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-1}\mathscr{F}\mathcal{A}.

Summing up the above all estimates gives

|u|Lϕφ4(u¯,u)|u0|Lϕφ4(u¯,u0)+(𝒜+~)(𝒜+~).less-than-or-similar-tosubscriptnorm𝑢𝐿italic-ϕ𝜑superscript4¯𝑢𝑢subscriptnormsubscript𝑢0𝐿italic-ϕ𝜑superscript4¯𝑢subscript𝑢0𝒜~less-than-or-similar-to𝒜~\||u|L\phi-\varphi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\||u_{0}|L\phi-\varphi\|_{\mathbb{H}^{4}(\underline{u},u_{0})}+\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{E}})\lesssim\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{E}}).

This also implies

|u|Lϕ4(u¯,u)(𝒜+~).less-than-or-similar-tosubscriptnorm𝑢𝐿italic-ϕsuperscript4¯𝑢𝑢𝒜~\||u|L\phi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{E}}). (4.16)
Remark 4.1.

~~\widetilde{\mathcal{E}} appears in the above inequality because we have not done anything to estimate the \engordnumber1 term on the right hand side but simply use the definition of ~~\widetilde{\mathcal{E}}. This term is also a borderline term although it is linear. But we will see from the proof that ~~\widetilde{\mathcal{E}} can be controlled without knowing (4.16).

Estimate for / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi: We use the equation

D/ ϕ=/ Lϕ.𝐷/ italic-ϕ/ 𝐿italic-ϕD\mbox{$\nabla\mkern-13.0mu/$ }\phi=\mbox{$\nabla\mkern-13.0mu/$ }L\phi. (4.17)

From this equation, we have

/ ϕ4(u¯,u)δ/ Lϕ𝕃u¯14(u)δ|u|2|u|(|u|/ )Lϕ𝕃u¯24(u)δ|u|2~.less-than-or-similar-tosubscriptdelimited-∥∥/ italic-ϕsuperscript4¯𝑢𝑢𝛿subscriptdelimited-∥∥/ 𝐿italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢less-than-or-similar-to𝛿superscript𝑢2subscriptdelimited-∥∥𝑢𝑢/ 𝐿italic-ϕsuperscriptsubscript𝕃¯𝑢2superscript4𝑢less-than-or-similar-to𝛿superscript𝑢2~\begin{split}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}&\lesssim\delta\|\mbox{$\nabla\mkern-13.0mu/$ }L\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim\delta|u|^{-2}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}_{\underline{u}}^{2}\mathbb{H}^{4}(u)}\lesssim\delta|u|^{-2}\mathscr{F}\mathscr{E}\widetilde{\mathcal{E}}.\end{split} (4.18)

Estimate for L¯ϕ¯𝐿italic-ϕ\underline{L}\phi: To estimate L¯ϕ¯𝐿italic-ϕ\underline{L}\phi, we use the equation for DL¯ϕ𝐷¯𝐿italic-ϕD\underline{L}\phi:

D(L¯ϕψ/|u|)+12ΩtrχL¯ϕ=Ω2Δ/ ϕ+2Ω2(η¯,/ ϕ)12Ωtrχ¯Lϕ.𝐷¯𝐿italic-ϕ𝜓𝑢12Ωtr𝜒¯𝐿italic-ϕsuperscriptΩ2Δ/ italic-ϕ2superscriptΩ2¯𝜂/ italic-ϕ12Ωtr¯𝜒𝐿italic-ϕD(\underline{L}\phi-\psi/|u|)+\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi=\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}(\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi. (4.19)

To estimate |u|L¯ϕψ𝑢¯𝐿italic-ϕ𝜓|u|\underline{L}\phi-\psi in 𝕃[u0,u]24(u¯)subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u}), we first estimate the right hand side in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)} and then integrate over [u0,u]subscript𝑢0𝑢[u_{0},u]. We remark that L¯ϕ¯𝐿italic-ϕ\underline{L}\phi cannot be estimated in 𝕃usubscriptsuperscript𝕃𝑢\mathbb{L}^{\infty}_{u} because ψ𝜓\psi can only be estimated in 𝕃u2subscriptsuperscript𝕃2𝑢\mathbb{L}^{2}_{u}. Now the right hand side can be estimated by, with the last term being estimated using (4.12) and (4.16),

δRHS𝕃u¯14(u)𝛿subscriptnormRHSsubscriptsuperscript𝕃1¯𝑢superscript4𝑢\displaystyle\delta\|\text{RHS}\|_{\mathbb{L}^{1}_{\underline{u}}{\mathbb{H}^{4}(u)}} Ω0δ32|u|52~+Ω02C12δ3|u|422𝒲12𝒜2+δ|u|2(𝒜+~)less-than-or-similar-toabsentsubscriptΩ0superscript𝛿32superscript𝑢52~superscriptsubscriptΩ02superscript𝐶12superscript𝛿3superscript𝑢4superscript2superscript2superscript𝒲12superscript𝒜2𝛿superscript𝑢2𝒜~\displaystyle\lesssim\Omega_{0}\delta^{\frac{3}{2}}|u|^{-\frac{5}{2}}\mathscr{F}\mathscr{E}\widetilde{\mathcal{E}}+\Omega_{0}^{2}C^{\frac{1}{2}}\delta^{3}|u|^{-4}\mathscr{F}^{2}\mathscr{E}^{2}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}+\delta|u|^{-2}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{E}})
δ|u|2(𝒜+~).less-than-or-similar-toabsent𝛿superscript𝑢2𝒜~\displaystyle\lesssim\delta|u|^{-2}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{E}}).

Multiplying the square of the above estimate by |u|52superscript𝑢52|u|^{\frac{5}{2}} and then integrating over [u0,u]subscript𝑢0𝑢[u_{0},u], we have

(u0u|u|52δRHS𝕃u¯14(u)2du)12δ|u|14(𝒜+~).less-than-or-similar-tosuperscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢52𝛿subscriptsuperscriptnormRHS2subscriptsuperscript𝕃1¯𝑢superscript4superscript𝑢differential-dsuperscript𝑢12𝛿superscript𝑢14𝒜~\left(\int_{u_{0}}^{u}|u^{\prime}|^{\frac{5}{2}}\delta\|\text{RHS}\|^{2}_{\mathbb{L}^{1}_{\underline{u}}{\mathbb{H}^{4}(u^{\prime})}}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\lesssim\delta|u|^{-\frac{1}{4}}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{E}}). (4.20)
Remark 4.2.

The last term on the right hand side is again a borderline term, the factor Lϕ𝐿italic-ϕL\phi in which cannot be estimated using the bootstrap assumptions (4.1). Instead, we should estimate Lϕ𝐿italic-ϕL\phi by (4.16) which we derived above. From now on, we will point it out when we are going to use an estimate which is derived previously. Otherwise, we will only use the bootstrap assumptions (4.1), or the estimates (4.10), (4.11), (4.12) and (4.13), or simply use the definitions of ~~\widetilde{\mathcal{E}}, 𝒪~~𝒪\widetilde{\mathcal{O}} and \mathcal{R}, even if we have derived the corresponding estimate previously.

The second term on the left hand side of (4.19) can be estimated by

δΩtrχL¯ϕ𝕃u¯14(u)less-than-or-similar-to𝛿subscriptnormΩtr𝜒¯𝐿italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢absent\displaystyle\delta\|\Omega\mathrm{tr}\chi\underline{L}\phi\|_{\mathbb{L}^{1}_{\underline{u}}{\mathbb{H}^{4}(u)}}\lesssim δΩtrχ𝕃u¯4(u)(|ψ||u|+L¯ϕψ/|u|𝕃u¯24(u))𝛿subscriptnormΩtr𝜒subscriptsuperscript𝕃¯𝑢superscript4𝑢𝜓𝑢subscriptnorm¯𝐿italic-ϕ𝜓𝑢subscriptsuperscript𝕃2¯𝑢superscript4𝑢\displaystyle\delta\|\Omega\mathrm{tr}\chi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}\left(\frac{|\psi|}{|u|}+\|\underline{L}\phi-\psi/|u|\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\right)
less-than-or-similar-to\displaystyle\lesssim δ|u|1(2Ω02h+C12δ|u|12𝒜2)(|ψ||u|+L¯ϕψ/|u|𝕃u¯24(u)).𝛿superscript𝑢12superscriptsubscriptΩ02superscript𝐶12𝛿superscript𝑢1superscript2superscript𝒜2𝜓𝑢subscriptnorm¯𝐿italic-ϕ𝜓𝑢subscriptsuperscript𝕃2¯𝑢superscript4𝑢\displaystyle\delta|u|^{-1}(2\Omega_{0}^{2}h+C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^{2}\mathcal{A}^{2})\left(\frac{|\psi|}{|u|}+\|\underline{L}\phi-\psi/|u|\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\right).

Multiply the square of the above estimate by |u|52superscript𝑢52|u|^{\frac{5}{2}} and then integrate over [u0,u]subscript𝑢0𝑢[u_{0},u], we have

(u0u|u|52δ2ΩtrχL¯ϕ𝕃u¯14(u)2du)12δ|u|14(u0uΩ04|ψ|2|u|du)12+C12δ2|u|542𝒜2(u0u|ψ|2|u|du)12+(1+C1𝒜)δ|u|1|u|34(|u|L¯ϕψ)𝕃u¯𝕃[u0,u]24δ|u|14𝒜less-than-or-similar-tosuperscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢52superscript𝛿2subscriptsuperscriptdelimited-∥∥Ωtr𝜒¯𝐿italic-ϕ2subscriptsuperscript𝕃1¯𝑢superscript4superscript𝑢differential-dsuperscript𝑢12𝛿superscript𝑢14superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ04superscript𝜓2superscript𝑢differential-dsuperscript𝑢12superscript𝐶12superscript𝛿2superscript𝑢54superscript2superscript𝒜2superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscript𝜓2superscript𝑢differential-dsuperscript𝑢121superscript𝐶1𝒜𝛿superscript𝑢1subscriptdelimited-∥∥superscript𝑢34𝑢¯𝐿italic-ϕ𝜓subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4less-than-or-similar-to𝛿superscript𝑢14𝒜\begin{split}&\left(\int_{u_{0}}^{u}|u^{\prime}|^{\frac{5}{2}}\delta^{2}\|\Omega\mathrm{tr}\chi\underline{L}\phi\|^{2}_{\mathbb{L}^{1}_{\underline{u}}{\mathbb{H}^{4}(u^{\prime})}}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\\ \lesssim&\delta|u|^{-\frac{1}{4}}\left(\int_{u_{0}}^{u}\frac{\Omega_{0}^{4}|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}+C^{\frac{1}{2}}\delta^{2}|u|^{-\frac{5}{4}}\mathscr{F}^{2}\mathcal{A}^{2}\left(\int_{u_{0}}^{u}\frac{|\psi|^{2}}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\\ &+(1+C^{-1}\mathscr{F}\mathcal{A})\delta|u|^{-1}\||u|^{\frac{3}{4}}(|u|\underline{L}\phi-\psi)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\\ \lesssim&\delta|u|^{-\frac{1}{4}}\mathscr{F}\mathcal{A}\end{split} (4.21)

Combining (4.20) and (4.21), we have

δ1|u|14|u|34(|u|L¯ϕψ)𝕃[u0,u]24(u¯)𝒜.less-than-or-similar-tosuperscript𝛿1superscript𝑢14subscriptnormsuperscript𝑢34𝑢¯𝐿italic-ϕ𝜓subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢𝒜\delta^{-1}|u|^{\frac{1}{4}}\||u|^{\frac{3}{4}}(|u|\underline{L}\phi-\psi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\mathscr{F}\mathcal{A}. (4.22)

The estimates (4.16), (4.18), (4.22) give

𝒜+~.less-than-or-similar-to𝒜~\displaystyle\mathcal{E}\lesssim\mathcal{A}+\widetilde{\mathcal{E}}.

 

4.4. Estimates for 𝒪𝒪\mathcal{O}

The next proposition is about the estimates for 𝒪𝒪\mathcal{O}, which is about the lower order derivatives of the connection coefficients:

Proposition 4.2.

Under the assumptions of Theorem 3.1 and the bootstrap assumptions (4.1), we have

𝒪𝒜+𝒪~[η,η¯]+~++𝒜1232.less-than-or-similar-to𝒪𝒜~𝒪𝜂¯𝜂~superscript𝒜12superscript32\mathcal{O}\lesssim\mathcal{A}+\widetilde{\mathcal{O}}[\eta,\underline{\eta}]+\widetilde{\mathcal{E}}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}.

Here 𝒪~[η,η]~𝒪𝜂𝜂\widetilde{\mathcal{O}}[\eta,\eta] means the norms of η,η¯𝜂¯𝜂\eta,\underline{\eta} in the definition of 𝒪~~𝒪\widetilde{\mathcal{O}}.

Proof.

Estimate for Ωχ^Ω^𝜒\Omega\widehat{\chi}: We consider the structure equation for D¯(Ωχ^)¯𝐷Ω^𝜒\underline{D}(\Omega\widehat{\chi}):

D¯^(Ωχ^)12Ωtrχ¯Ωχ^=Ω2(/ ^η+η^η12trχχ¯^+/ ϕ^/ ϕ).^¯𝐷Ω^𝜒12Ωtr¯𝜒Ω^𝜒superscriptΩ2/ ^tensor-product𝜂𝜂^tensor-product𝜂12tr𝜒^¯𝜒/ italic-ϕ^tensor-product/ italic-ϕ\widehat{\underline{D}}(\Omega\widehat{\chi})-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\Omega\widehat{\chi}=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\eta+\eta\widehat{\otimes}\eta-\frac{1}{2}\mathrm{tr}\chi\widehat{\underline{\chi}}+\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi). (4.23)

The right hand side should be estimated in |u|2𝕃[u0,u]14(u¯)\||u|^{2}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}. The \engordnumber1, \engordnumber2 and \engordnumber4 term (denoted by I𝐼I) on the right hand side can be estimated by

|u|2I𝕃[u0,u]14(u¯)less-than-or-similar-tosubscriptnormsuperscript𝑢2𝐼subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢absent\displaystyle\||u|^{2}I\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim Ω0(u0)|u|12|u|32Ω0η𝕃[u0,u]25(u¯)+|u|2𝕃[u0,u]1|u|2(η,/ ϕ)𝕃[u0,u]4(u¯)2subscriptΩ0subscript𝑢0superscript𝑢12subscriptnormsuperscript𝑢32subscriptΩ0𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢subscriptnormsuperscript𝑢2subscriptsuperscript𝕃1subscript𝑢0𝑢superscriptsubscriptnormsuperscript𝑢2𝜂/ italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢2\displaystyle\Omega_{0}(u_{0})|u|^{-\frac{1}{2}}\||u|^{\frac{3}{2}}\Omega_{0}\eta\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}+\||u|^{-2}\|_{\mathbb{L}^{1}_{[u_{0},u]}}\||u|^{2}(\eta,\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}
less-than-or-similar-to\displaystyle\lesssim Ω0(u0)δ12|u|12(𝒜+𝒪~[η])+Ω02(u0)C12δ2|u|222𝒜2subscriptΩ0subscript𝑢0superscript𝛿12superscript𝑢12𝒜~𝒪delimited-[]𝜂superscriptsubscriptΩ02subscript𝑢0superscript𝐶12superscript𝛿2superscript𝑢2superscript2superscript2superscript𝒜2\displaystyle\Omega_{0}(u_{0})\delta^{\frac{1}{2}}|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta])+\Omega_{0}^{2}(u_{0})C^{\frac{1}{2}}\delta^{2}|u|^{-2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}
less-than-or-similar-to\displaystyle\lesssim (𝒜+𝒪~[η]).𝒜~𝒪delimited-[]𝜂\displaystyle\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta]).

The \engordnumber3 term on the right hand side is estimated by

|u|2ΩtrχΩχ¯^𝕃[u0,u]14(u¯)|u|1𝒜C14δ𝒜C1𝒜.less-than-or-similar-toevaluated-atnormsuperscript𝑢2Ωtr𝜒Ω^¯𝜒subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢superscript𝑢1𝒜superscript𝐶14𝛿𝒜less-than-or-similar-tosuperscript𝐶1𝒜\displaystyle\||u|^{2}\Omega\mathrm{tr}\chi\Omega\widehat{\underline{\chi}}\|\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\lesssim C^{-1}\mathscr{F}\mathcal{A}.

Therefore we have

|u|Ωχ^4(u¯,u)|u0|Ωχ^4(u¯,u0)+(𝒜+𝒪~[η])(𝒜+𝒪~[η]).less-than-or-similar-to𝑢subscriptnormΩ^𝜒superscript4¯𝑢𝑢subscript𝑢0subscriptnormΩ^𝜒superscript4¯𝑢subscript𝑢0𝒜~𝒪delimited-[]𝜂less-than-or-similar-to𝒜~𝒪delimited-[]𝜂\displaystyle|u|\|\Omega\widehat{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim|u_{0}|\|\Omega\widehat{\chi}\|_{\mathbb{H}^{4}(\underline{u},u_{0})}+\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta])\lesssim\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta]). (4.24)

Estimate for trχtrsuperscript𝜒\mathrm{tr}\chi^{\prime}: We estimate trχtrsuperscript𝜒\mathrm{tr}\chi^{\prime} by the equation for Dtrχ𝐷trsuperscript𝜒D\mathrm{tr}\chi^{\prime} written in form:

D(trχ2h|u|)=Ω2(12(Ω2trχ)2|Ωχ^|22(Lϕ)2).𝐷trsuperscript𝜒2𝑢superscriptΩ212superscriptsuperscriptΩ2trsuperscript𝜒2superscriptΩ^𝜒22superscript𝐿italic-ϕ2D\left(\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|}\right)=\Omega^{-2}\left(-\frac{1}{2}(\Omega^{2}\mathrm{tr}\chi^{\prime})^{2}-|\Omega\widehat{\chi}|^{2}-2(L\phi)^{2}\right). (4.25)

We then have (using (4.11), (4.16) and (4.24))

trχ2h|u|4(u¯,u)less-than-or-similar-tosubscriptnormtrsuperscript𝜒2𝑢superscript4¯𝑢𝑢absent\displaystyle\left\|\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|}\right\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim Ω02δ|u|22(𝒜2+~2+𝒪~[η]2).superscriptsubscriptΩ02𝛿superscript𝑢2superscript2superscript𝒜2superscript~2~𝒪superscriptdelimited-[]𝜂2\displaystyle\Omega_{0}^{-2}\delta|u|^{-2}\mathscr{F}^{2}(\mathcal{A}^{2}+\widetilde{\mathcal{E}}^{2}+\widetilde{\mathcal{O}}[\eta]^{2}). (4.26)

Estimate for η𝜂\eta: We write the equation for Dη𝐷𝜂D\eta in the following form:

Dη=(Ωχ)η¯2Lϕ/ ϕ(ΩβLϕ/ ϕ).𝐷𝜂Ω𝜒¯𝜂2𝐿italic-ϕ/ italic-ϕΩ𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle D\eta=(\Omega\chi)\cdot\underline{\eta}-2L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi-(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi). (4.27)

The first two terms on the right hand side can be estimated in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)} by

δ|u|1C14𝒜C14δ|u|2𝒲12𝒜C1δ|u|2𝒜.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜less-than-or-similar-tosuperscript𝐶1𝛿superscript𝑢2𝒜\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-1}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathcal{A}.

The last term on the right hand side can be estimated in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)} by

δΩβLϕ/ ϕ𝕃u¯24(u)δ|u|2.less-than-or-similar-toabsent𝛿subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢less-than-or-similar-to𝛿superscript𝑢2\displaystyle\lesssim\delta\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathcal{R}.

We have

η4(u¯,u)δ|u|2(C1𝒜+)δ|u|2(𝒜+).less-than-or-similar-tosubscriptnorm𝜂superscript4¯𝑢𝑢𝛿superscript𝑢2superscript𝐶1𝒜less-than-or-similar-to𝛿superscript𝑢2𝒜\|\eta\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\delta|u|^{-2}\mathscr{F}\mathscr{E}(C^{-1}\mathcal{A}+\mathcal{R})\lesssim\delta|u|^{-2}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}). (4.28)

Estimate for η¯¯𝜂\underline{\eta}: We then write the equation for D¯η¯¯𝐷¯𝜂\underline{D}\underline{\eta} in the following form:

D¯η¯=(Ωχ¯)η2L¯ϕ/ ϕ+(Ωβ¯+L¯ϕ/ ϕ).¯𝐷¯𝜂Ω¯𝜒𝜂2¯𝐿italic-ϕ/ italic-ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\underline{D}\underline{\eta}=(\Omega\underline{\chi})\cdot\eta-2\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi). (4.29)

The right hand side should be estimated in |u|2𝕃[u0,u]14(u¯)\||u|^{2}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}. Note that

Ωχ¯4(u¯,u)Ωtrχ¯4(u¯,u)+Ωχ¯^4(u¯,u)|u|1+C14δ|u|2𝒜|u|1.less-than-or-similar-tosubscriptnormΩ¯𝜒superscript4¯𝑢𝑢subscriptnormΩtr¯𝜒superscript4¯𝑢𝑢subscriptnormΩ^¯𝜒superscript4¯𝑢𝑢less-than-or-similar-tosuperscript𝑢1superscript𝐶14𝛿superscript𝑢2𝒜less-than-or-similar-tosuperscript𝑢1\displaystyle\|\Omega\underline{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\|\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}+\|\Omega\widehat{\underline{\chi}}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim|u|^{-1}+C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathcal{A}\lesssim|u|^{-1}.

Using (4.28), the \engordnumber1 term on the right hand side is estimated by

|u|2Ωχ¯η𝕃[u0,u]14(u¯)δ|u|1(𝒜+).less-than-or-similar-tosubscriptnormsuperscript𝑢2Ω¯𝜒𝜂subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢𝛿superscript𝑢1𝒜\displaystyle\||u|^{2}\Omega\underline{\chi}\eta\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\delta|u|^{-1}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}).

Using (4.13) and (4.18), the \engordnumber2 term is estimated by

|u|2L¯ϕ/ ϕ𝕃[u0,u]14(u¯)less-than-or-similar-tosubscriptnormsuperscript𝑢2¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢absent\displaystyle\||u|^{2}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim |u|L¯ϕ𝕃[u0,u]24(u¯)|u|/ ϕ𝕃[u0,u]24(u¯)subscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢subscriptnorm𝑢/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢\displaystyle\||u|\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\cdot\||u|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim δ|u|1𝒲12~.𝛿superscript𝑢1superscript𝒲12~\displaystyle\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\widetilde{\mathcal{E}}.

The last term is estimated by

|u|2(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]14(u¯)less-than-or-similar-tosubscriptnormsuperscript𝑢2Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢absent\displaystyle\||u|^{2}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim Ω0|u|32𝕃[u0,u]24(u¯)Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]24(u¯)subscriptnormsubscriptΩ0superscript𝑢32subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢subscriptnormsuperscriptsubscriptΩ01superscript𝑢72Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢\displaystyle\|\Omega_{0}|u|^{-\frac{3}{2}}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\cdot\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim Ω0(u0)Ω01δ32|u|323232δ|u|1𝒜1232.less-than-or-similar-tosubscriptΩ0subscript𝑢0superscriptsubscriptΩ01superscript𝛿32superscript𝑢32superscript32superscript32𝛿superscript𝑢1superscript𝒜12superscript32\displaystyle\Omega_{0}(u_{0})\Omega_{0}^{-1}\delta^{\frac{3}{2}}|u|^{-\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{R}^{\frac{3}{2}}\lesssim\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}.

where we have used the auxiliary condition (3.5). Combining all estimates above, we have

|u|η¯4(u¯,u)|u0|η¯4(u¯,u0)+δ|u|1𝒲12(𝒜+𝒜1232+~)δ|u|1𝒲12(𝒜+𝒜1232+~).less-than-or-similar-tosubscriptdelimited-∥∥𝑢¯𝜂superscript4¯𝑢𝑢subscriptdelimited-∥∥subscript𝑢0¯𝜂superscript4¯𝑢subscript𝑢0𝛿superscript𝑢1superscript𝒲12𝒜superscript𝒜12superscript32~less-than-or-similar-to𝛿superscript𝑢1superscript𝒲12𝒜superscript𝒜12superscript32~\begin{split}\||u|\underline{\eta}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim&\||u_{0}|\underline{\eta}\|_{\mathbb{H}^{4}(\underline{u},u_{0})}+\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}+\widetilde{\mathcal{E}})\\ \lesssim&\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}+\widetilde{\mathcal{E}}).\end{split} (4.30)

Estimate for Ωχ¯^Ω^¯𝜒\Omega\widehat{\underline{\chi}}: Remember the equation for D(Ωχ¯^)𝐷Ω^¯𝜒D(\Omega\widehat{\underline{\chi}}):

D^(Ωχ¯^)=Ω2(/ ^η¯+η¯^η¯+/ ϕ^/ ϕ)+12ΩtrχΩχ¯^12Ωtrχ¯Ωχ^.^𝐷Ω^¯𝜒superscriptΩ2/ ^tensor-product¯𝜂¯𝜂^tensor-product¯𝜂/ italic-ϕ^tensor-product/ italic-ϕ12Ωtr𝜒Ω^¯𝜒12Ωtr¯𝜒Ω^𝜒\displaystyle\widehat{D}(\Omega\widehat{\underline{\chi}})=\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\underline{\eta}+\underline{\eta}\widehat{\otimes}\underline{\eta}+\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\frac{1}{2}\Omega\mathrm{tr}\chi\Omega\widehat{\underline{\chi}}-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\Omega\widehat{\chi}. (4.31)

The first three terms on the right hand side are estimated in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{4}(u)} by

Ω0(u0)δ32|u|52𝒲12𝒪~[η¯]+Ω02(u0)C12δ3|u|422𝒲𝒜2δ|u|2(𝒜+𝒪~[η¯]).less-than-or-similar-toabsentsubscriptΩ0subscript𝑢0superscript𝛿32superscript𝑢52superscript𝒲12~𝒪delimited-[]¯𝜂superscriptsubscriptΩ02subscript𝑢0superscript𝐶12superscript𝛿3superscript𝑢4superscript2superscript2𝒲superscript𝒜2less-than-or-similar-to𝛿superscript𝑢2𝒜~𝒪delimited-[]¯𝜂\displaystyle\lesssim\Omega_{0}(u_{0})\delta^{\frac{3}{2}}|u|^{-\frac{5}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\widetilde{\mathcal{O}}[\underline{\eta}]+\Omega_{0}^{2}(u_{0})C^{\frac{1}{2}}\delta^{3}|u|^{-4}\mathscr{F}^{2}\mathscr{E}^{2}\mathscr{W}\mathcal{A}^{2}\lesssim\delta|u|^{-2}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\underline{\eta}]). (4.32)

In the same norm, the \engordnumber4 term is estimated by δ|u|1𝒜C14δ|u|2𝒜C1δ|u|2𝒜less-than-or-similar-toabsent𝛿superscript𝑢1𝒜superscript𝐶14𝛿superscript𝑢2𝒜less-than-or-similar-tosuperscript𝐶1𝛿superscript𝑢2𝒜\lesssim\delta\cdot|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathcal{A}\lesssim C^{-1}\delta|u|^{-2}\mathscr{F}\mathcal{A}. The \engordnumber5 term is estimated by (using (4.12) and (4.24)) δ|u|2(𝒜+𝒪~[η])less-than-or-similar-toabsent𝛿superscript𝑢2𝒜~𝒪delimited-[]𝜂\lesssim\delta|u|^{-2}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta]). Therefore, we have

Ωχ¯^4(u¯,u)δ|u|2(𝒜+𝒪~[η,η¯]).less-than-or-similar-tosubscriptnormΩ^¯𝜒superscript4¯𝑢𝑢𝛿superscript𝑢2𝒜~𝒪𝜂¯𝜂\displaystyle\|\Omega\widehat{\underline{\chi}}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\delta|u|^{-2}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta,\underline{\eta}]). (4.33)

Estimate for Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi}: Write the equation for D(Ωtrχ¯)𝐷Ωtr¯𝜒D(\Omega\mathrm{tr}\underline{\chi}):

D(Ωtrχ¯+2|u|)=Ω2(2div/ η¯+2|η¯|2+2|/ ϕ|22K)ΩtrχΩtrχ¯.𝐷Ωtr¯𝜒2𝑢superscriptΩ22div/ ¯𝜂2superscript¯𝜂22superscript/ italic-ϕ22𝐾Ωtr𝜒Ωtr¯𝜒D\left(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\right)=\Omega^{2}(2\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}+2|\underline{\eta}|^{2}+2|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}-2K)-\Omega\mathrm{tr}\chi\Omega\mathrm{tr}\underline{\chi}. (4.34)

The right hand side is estimated in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}_{\underline{u}}^{1}\mathbb{H}^{4}(u)}. The estimates for the first three terms are the same with (4.32). The last term is estimated by δ|u|2𝒜less-than-or-similar-toabsent𝛿superscript𝑢2𝒜\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}. The Gauss curvature term Ω2KsuperscriptΩ2𝐾\Omega^{2}K is estimated by

δΩ2K𝕃u¯14(u)δΩ02K|u|2𝕃u¯24(u)+δ|u|2Ω0C38δ2|u|332𝒜32+δ|u|2δ|u|2𝒜.less-than-or-similar-to𝛿subscriptdelimited-∥∥superscriptΩ2𝐾subscriptsuperscript𝕃1¯𝑢superscript4𝑢𝛿superscriptsubscriptΩ02subscriptdelimited-∥∥𝐾superscript𝑢2subscriptsuperscript𝕃2¯𝑢superscript4𝑢𝛿superscript𝑢2less-than-or-similar-tosubscriptΩ0superscript𝐶38superscript𝛿2superscript𝑢3superscript32superscript𝒜32𝛿superscript𝑢2less-than-or-similar-to𝛿superscript𝑢2𝒜\begin{split}\delta\|\Omega^{2}K\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim&\delta\Omega_{0}^{2}\|K-|u|^{-2}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta|u|^{-2}\\ \lesssim&\Omega_{0}C^{\frac{3}{8}}\delta^{2}|u|^{-3}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}+\delta|u|^{-2}\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}.\end{split} (4.35)

Combining the estimates above we have

Ωtrχ¯+2|u|14(u¯,u)δ|u|2(𝒜+𝒪~[η]).less-than-or-similar-tosubscriptnormΩtr¯𝜒2superscript𝑢1superscript4¯𝑢𝑢𝛿superscript𝑢2𝒜~𝒪delimited-[]𝜂\|\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\delta|u|^{-2}\mathscr{F}(\mathcal{A}+\widetilde{\mathcal{O}}[\eta]). (4.36)

Improved estimate for the derivatives of Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi}: We should mention that the derivatives of Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} behave better from the perspective of the power of δ|u|1𝛿superscript𝑢1\delta|u|^{-1}. This is because the derivatives of Ωtrχ,Ωtrχ¯Ωtr𝜒Ωtr¯𝜒\Omega\mathrm{tr}\chi,\Omega\mathrm{tr}\underline{\chi} and K𝐾K behave better than themselves. Note that

δ(|u|/ )(2Ω2KΩtrχΩtrχ¯)𝕃u¯13(u)𝛿subscriptnorm𝑢/ 2superscriptΩ2𝐾Ωtr𝜒Ωtr¯𝜒subscriptsuperscript𝕃1¯𝑢superscript3𝑢\displaystyle\delta\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })(-2\Omega^{2}K-\Omega\mathrm{tr}\chi\Omega\mathrm{tr}\underline{\chi})\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{3}(u)}
less-than-or-similar-to\displaystyle\lesssim δΩ02K|u|2𝕃u¯24(u)+δΩ02trχ2h|u|1𝕃u¯4(u¯,u)Ωtrχ¯+2|u|1𝕃u¯4(u¯,u)𝛿superscriptsubscriptΩ02subscriptnorm𝐾superscript𝑢2subscriptsuperscript𝕃2¯𝑢superscript4𝑢𝛿superscriptsubscriptΩ02subscriptnormtrsuperscript𝜒2superscript𝑢1subscriptsuperscript𝕃¯𝑢superscript4¯𝑢𝑢subscriptnormΩtr¯𝜒2superscript𝑢1subscriptsuperscript𝕃¯𝑢superscript4¯𝑢𝑢\displaystyle\delta\Omega_{0}^{2}\|K-|u|^{-2}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta\Omega_{0}^{2}\|\mathrm{tr}\chi^{\prime}-2h|u|^{-1}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(\underline{u},u)}\|\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(\underline{u},u)}
+δΩ02(Ωtrχ¯𝕃u¯𝕃(u)trχ2h|u|1𝕃u¯4(u¯,u)+trχ𝕃u¯𝕃(u)Ωtrχ¯+2|u|1𝕃u¯4(u¯,u))𝛿superscriptsubscriptΩ02subscriptnormΩtr¯𝜒subscriptsuperscript𝕃¯𝑢superscript𝕃𝑢subscriptnormtrsuperscript𝜒2superscript𝑢1subscriptsuperscript𝕃¯𝑢superscript4¯𝑢𝑢subscriptnormtrsuperscript𝜒subscriptsuperscript𝕃¯𝑢superscript𝕃𝑢subscriptnormΩtr¯𝜒2superscript𝑢1subscriptsuperscript𝕃¯𝑢superscript4¯𝑢𝑢\displaystyle+\delta\Omega_{0}^{2}\left(\|\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}(u)}\|\mathrm{tr}\chi^{\prime}-2h|u|^{-1}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(\underline{u},u)}+\|\mathrm{tr}\chi^{\prime}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}(u)}\|\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(\underline{u},u)}\right)
less-than-or-similar-to\displaystyle\lesssim Ω0C38δ2|u|332𝒜32+C34δ3|u|43𝒜3+C14δ2|u|32𝒜2subscriptΩ0superscript𝐶38superscript𝛿2superscript𝑢3superscript32superscript𝒜32superscript𝐶34superscript𝛿3superscript𝑢4superscript3superscript𝒜3superscript𝐶14superscript𝛿2superscript𝑢3superscript2superscript𝒜2\displaystyle\Omega_{0}C^{\frac{3}{8}}\delta^{2}|u|^{-3}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}+C^{\frac{3}{4}}\delta^{3}|u|^{-4}\mathscr{F}^{3}\mathcal{A}^{3}+C^{\frac{1}{4}}\delta^{2}|u|^{-3}\mathscr{F}^{2}\mathcal{A}^{2}

Combining the above estimates with the estimates for the first three terms, which are bounded by C14Ω0δ32|u|52𝒲12𝒜superscript𝐶14subscriptΩ0superscript𝛿32superscript𝑢52superscript𝒲12𝒜C^{\frac{1}{4}}\Omega_{0}\delta^{\frac{3}{2}}|u|^{-\frac{5}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A} in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}, gives

(|u|/ )(Ωtrχ¯)3(u¯,u)C14δ32|u|52𝒲12𝒜.less-than-or-similar-tosubscriptnorm𝑢/ Ωtr¯𝜒superscript3¯𝑢𝑢superscript𝐶14superscript𝛿32superscript𝑢52superscript𝒲12𝒜\displaystyle\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim C^{\frac{1}{4}}\delta^{\frac{3}{2}}|u|^{-\frac{5}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (4.37)

Estimate for ω𝜔\omega: Consider the equation

D¯ω=Ω2(2(η,η¯)|η¯|2|/ ϕ|2+K)+14ΩtrχΩtrχ¯12(Ωχ^,Ωχ¯^)+LϕL¯ϕ,¯𝐷𝜔superscriptΩ22𝜂¯𝜂superscript¯𝜂2superscript/ italic-ϕ2𝐾14Ωtr𝜒Ωtr¯𝜒12Ω^𝜒Ω^¯𝜒𝐿italic-ϕ¯𝐿italic-ϕ\underline{D}\omega=\Omega^{2}(2(\eta,\underline{\eta})-|\underline{\eta}|^{2}-|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}+K)+\frac{1}{4}\Omega\mathrm{tr}\chi\Omega\mathrm{tr}\underline{\chi}-\frac{1}{2}(\Omega\widehat{\chi},\Omega\widehat{\underline{\chi}})+L\phi\underline{L}\phi, (4.38)

The right hand side should be estimated in |u|𝕃[u0,u]14(u¯)\||u|\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}. The first three terms are estimated by

Ω02(u0)C12δ2|u|322𝒲𝒜2C2|u|1𝒜.less-than-or-similar-toabsentsuperscriptsubscriptΩ02subscript𝑢0superscript𝐶12superscript𝛿2superscript𝑢3superscript2superscript2𝒲superscript𝒜2less-than-or-similar-tosuperscript𝐶2superscript𝑢1𝒜\displaystyle\lesssim\Omega_{0}^{2}(u_{0})C^{\frac{1}{2}}\delta^{2}|u|^{-3}\mathscr{F}^{2}\mathscr{E}^{2}\mathscr{W}\mathcal{A}^{2}\lesssim C^{-2}|u|^{-1}\mathscr{F}\mathcal{A}.

The last three terms are estimated by, using (4.16),

|u|1𝒲12(𝒜+~).less-than-or-similar-toabsentsuperscript𝑢1superscript𝒲12𝒜~\displaystyle\lesssim|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\widetilde{\mathcal{E}}).

The Gauss curvature term Ω2KsuperscriptΩ2𝐾\Omega^{2}K is estimated by

|u|2Ω2K𝕃[u0,u]14(u¯)less-than-or-similar-tosubscriptnormsuperscript𝑢2superscriptΩ2𝐾subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢absent\displaystyle\||u|^{2}\Omega^{2}K\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim |u|1+Ω0|u|32Ω0|u|52(K|u|2)𝕃[u0,u]24(u¯)superscript𝑢1subscriptΩ0superscript𝑢32subscriptnormsubscriptΩ0superscript𝑢52𝐾superscript𝑢2subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢\displaystyle|u|^{-1}+\Omega_{0}|u|^{-\frac{3}{2}}\|\Omega_{0}|u|^{\frac{5}{2}}(K-|u|^{-2})\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim |u|1+Ω0δ12|u|32(𝒜+)superscript𝑢1subscriptΩ0superscript𝛿12superscript𝑢32𝒜\displaystyle|u|^{-1}+\Omega_{0}\delta^{\frac{1}{2}}|u|^{-\frac{3}{2}}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R})
less-than-or-similar-to\displaystyle\lesssim |u|1(𝒜+).superscript𝑢1𝒜\displaystyle|u|^{-1}\mathscr{F}(\mathcal{A}+\mathcal{R}).

Combining the above estimates gives

ω4(u¯,u)|u|1𝒲12(𝒜+~+).less-than-or-similar-tosubscriptnorm𝜔superscript4¯𝑢𝑢superscript𝑢1superscript𝒲12𝒜~\displaystyle\|\omega\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\widetilde{\mathcal{E}}+\mathcal{R}). (4.39)

The estimates (4.24), (4.26), (4.28), (4.30), (4.33), (4.36), (4.39) are the desired estimates and the proof is completed.  

4.5. Estimates for ~~\widetilde{\mathcal{E}}

We then turn to ~~\widetilde{\mathcal{E}}, which is about the derivatives of the derivatives of the wave function ϕitalic-ϕ\phi.

Proposition 4.3.

Under the assumptions of Theorem 3.1 and the bootstrap assumptions (4.1), we have

~𝒜.less-than-or-similar-to~𝒜\widetilde{\mathcal{E}}\lesssim\mathcal{A}.
Proof.

Recall the following equations which are essentially the wave equation:

D¯Lϕ+12Ωtrχ¯Lϕ=Ω2div/ / ϕ+2Ω2(η,/ ϕ)12ΩtrχL¯ϕ,D/ ϕ=/ Lϕ,DL¯ϕ+12ΩtrχL¯ϕ=Ω2div/ / ϕ+2Ω2(η¯,/ ϕ)12Ωtrχ¯Lϕ,D¯/ ϕ=/ L¯ϕ.formulae-sequence¯𝐷𝐿italic-ϕ12Ωtr¯𝜒𝐿italic-ϕsuperscriptΩ2div/ / italic-ϕ2superscriptΩ2𝜂/ italic-ϕ12Ωtr𝜒¯𝐿italic-ϕformulae-sequence𝐷/ italic-ϕ/ 𝐿italic-ϕformulae-sequence𝐷¯𝐿italic-ϕ12Ωtr𝜒¯𝐿italic-ϕsuperscriptΩ2div/ / italic-ϕ2superscriptΩ2¯𝜂/ italic-ϕ12Ωtr¯𝜒𝐿italic-ϕ¯𝐷/ italic-ϕ/ ¯𝐿italic-ϕ\begin{split}\underline{D}L\phi+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi&=\Omega^{2}\mbox{$\mathrm{div}\mkern-13.0mu/$ }\mbox{$\nabla\mkern-13.0mu/$ }\phi+2\Omega^{2}(\eta,\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi,\\ D\mbox{$\nabla\mkern-13.0mu/$ }\phi&=\mbox{$\nabla\mkern-13.0mu/$ }L\phi,\\ D\underline{L}\phi+\frac{1}{2}\Omega\mathrm{tr}\chi\underline{L}\phi&=\Omega^{2}\mbox{$\mathrm{div}\mkern-13.0mu/$ }\mbox{$\nabla\mkern-13.0mu/$ }\phi+2\Omega^{2}(\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}L\phi,\\ \underline{D}\mbox{$\nabla\mkern-13.0mu/$ }\phi&=\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi.\end{split} (4.40)

We compute

D¯(|(|u|/ )iLϕ|2dμg/)+D(Ω2|(|u|/ )i/ ϕ|2dμg/)\displaystyle\underline{D}(|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}L\phi|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})+D(\Omega^{2}|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})
=\displaystyle= |u|2i/ A(Ω2/ B1B5i/ Aϕ/ i,B1B5Lϕ)+τ1superscript𝑢2𝑖superscript/ 𝐴superscriptΩ2subscriptsuperscript/ 𝑖subscript𝐵1subscript𝐵5subscript/ 𝐴italic-ϕsuperscript/ 𝑖subscript𝐵1subscript𝐵5𝐿italic-ϕsubscript𝜏1\displaystyle|u|^{2i}\mbox{$\nabla\mkern-13.0mu/$ }^{A}(\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }^{i}_{B_{1}\cdots B_{5}}\mbox{$\nabla\mkern-13.0mu/$ }_{A}\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }^{i,B_{1}\cdots B_{5}}L\phi)+\tau_{1}

for 1i51𝑖51\leq i\leq 5, where τ1subscript𝜏1\tau_{1} contains no (i+1)stsuperscript𝑖1st(i+1)^{\text{st}} order derivatives of the derivative of ϕitalic-ϕ\phi. By divergence theorem, we obtain

δ|u|(|u|/ )Lϕ𝕃u¯24(u)2+|u|32Ω(|u|/ )/ ϕ𝕃[u0,u]24(u¯)2𝛿superscriptsubscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢2superscriptsubscriptnormsuperscript𝑢32Ω𝑢/ / italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢2\displaystyle\delta\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}+\||u|^{\frac{3}{2}}\Omega(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}
less-than-or-similar-to\displaystyle\lesssim δ|u|(|u|/ )Lϕ𝕃u¯24(u0)2+|u|32Ω(|u|/ )/ ϕ𝕃[u0,u]24(0)2+0δdu¯u0uduSu¯,u|τ1|dμg/\displaystyle\delta\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\||u|^{\frac{3}{2}}\Omega(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(0)}^{2}+\int_{0}^{\delta}\mathrm{d}\underline{u}^{\prime}\int_{u_{0}}^{u}\mathrm{d}u^{\prime}\int_{S_{\underline{u}^{\prime},u^{\prime}}}|\tau_{1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

We remark that the power 111 of the multiple |u|𝑢|u| before (|u|/ )Lϕ𝑢/ 𝐿italic-ϕ(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi above comes in because the coefficient of Ωtrχ¯LϕΩtr¯𝜒𝐿italic-ϕ\Omega\mathrm{tr}\underline{\chi}L\phi on the left hand side is 1212\frac{1}{2}, and then Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} itself will not appear in the following estimates (instead, only the normalized quantity Ωtrχ¯+2|u|1Ωtr¯𝜒2superscript𝑢1\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1} comes in). And by direct computation,

Su¯,u|τ1|dμg/Su¯,u|τ1,1|dμg/+Su¯,u|τ1,2|dμg/\displaystyle\int_{S_{\underline{u}^{\prime},u^{\prime}}}|\tau_{1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim\int_{S_{\underline{u}^{\prime},u^{\prime}}}|\tau_{1,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}+\int_{S_{\underline{u}^{\prime},u^{\prime}}}|\tau_{1,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

where (the norm i\|\cdot\|_{i} refers to i(u¯,u)\|\cdot\|_{\mathbb{H}^{i}(\underline{u}^{\prime},u^{\prime})})

Su¯,u|τ1,1|dμg/|u|2(|u|/ )Lϕ4(Ωtrχ¯+2|u|25Lϕ4+Ωtrχ¯+2|u|24(|u|/ )Lϕ4)+|u|2(|u|/ )Ω24(|u|1(|u|/ )/ ϕ4)(|u|/ )Lϕ4+|u|2(|u|/ )Lϕ4(Ω25η4/ ϕ4+Ω24η5/ ϕ4+Ω24η4/ ϕ5)+|u|2(|u|/ )Lϕ4(Ωtrχ4(|u|/ )L¯ϕ4+(|u|/ )(Ωtrχ)4L¯ϕ4)+|u|2Ω02(u)|ω|(|u|/ )/ ϕ42\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}|\tau_{1,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&|u^{\prime}|^{2}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4}(\|\Omega\mathrm{tr}\underline{\chi}+2|u^{\prime}|^{-2}\|_{5}\|L\phi\|_{4}+\|\Omega\mathrm{tr}\underline{\chi}+2|u^{\prime}|^{-2}\|_{4}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4})\\ &+|u^{\prime}|^{2}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\Omega^{2}\|_{4}(|u^{\prime}|^{-1}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4})\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4}\\ &+|u^{\prime}|^{2}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4}\left(\|\Omega^{2}\|_{5}\|\eta\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+\|\Omega^{2}\|_{4}\|\eta\|_{5}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+\|\Omega^{2}\|_{4}\|\eta\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{5}\right)\\ &+|u^{\prime}|^{2}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4}\left(\|\Omega\mathrm{tr}\chi\|_{4}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}+\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{4}\|\underline{L}\phi\|_{4}\right)\\ &+|u^{\prime}|^{2}\Omega_{0}^{2}(u^{\prime})|\omega|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}^{2}\end{split} (4.41)

which we call the multiplier terms, and

Su¯,u|τ1,2|dμg/|u|2(|u|/ )Lϕ4((|u|/ )(Ωχ¯)3Lϕ4+Ω02(u)|u|K4/ ϕ4)+|u|2Ω02(u)(|u|/ )/ ϕ4((|u|/ )(Ωχ)4/ ϕ4+|u|K3Lϕ4)\begin{split}\int_{S_{\underline{u}^{\prime},u^{\prime}}}|\tau_{1,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim&|u^{\prime}|^{2}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\underline{\chi})\|_{3}\|L\phi\|_{4}+\Omega_{0}^{2}(u^{\prime})|u^{\prime}|\|K\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\right)\\ &+|u^{\prime}|^{2}\Omega_{0}^{2}(u^{\prime})\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\chi)\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+|u^{\prime}|\|K\|_{3}\|L\phi\|_{4}\right)\end{split} (4.42)

which we call the commutation terms. We will estimate these terms in Lu¯1Lu1subscriptsuperscript𝐿1¯𝑢subscriptsuperscript𝐿1𝑢L^{1}_{\underline{u}}L^{1}_{u}. The \engordnumber1 term of the \engordnumber1 line on the right hand side of (4.41) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|(|u|/ )Lϕ𝕃u¯24(u)|u|2(Ωtrχ¯+2|u|1)𝕃u¯𝕃[u0,u]5|u|Lϕ𝕃u¯𝕃[u0,u]4𝛿superscript𝑢1subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢subscriptnormsuperscript𝑢2Ωtr¯𝜒2superscript𝑢1subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript5subscriptnorm𝑢𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-1}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\||u|^{2}(\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1})\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{5}}\||u|L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C14𝒜C14δ𝒲12𝒜C14𝒜C14δ22𝒜2(δ|u|1𝒜)12.less-than-or-similar-to𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14𝒜superscript𝐶14𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒜12\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\lesssim C^{-\frac{1}{4}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{2}}.

The \engordnumber2 term of the \engordnumber1 line of (4.41) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|(|u|/ )Lϕ𝕃u¯24(u)2|u|2(Ωtrχ¯+2|u|1)𝕃u¯𝕃[u0,u]4𝛿superscript𝑢1subscriptsuperscriptnorm𝑢𝑢/ 𝐿italic-ϕ2subscriptsuperscript𝕃2¯𝑢superscript4𝑢subscriptnormsuperscript𝑢2Ωtr¯𝜒2superscript𝑢1subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-1}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|^{2}_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\||u|^{2}(\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1})\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C1222𝒜2C14δ𝒜C14δ22𝒜2(δ|u|1𝒜)12.less-than-or-similar-to𝛿superscript𝑢1superscript𝐶12superscript2superscript2superscript𝒜2superscript𝐶14𝛿𝒜superscript𝐶14𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒜12\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{2}}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\lesssim C^{-\frac{1}{4}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{2}}.

Noting that

(|u|/ )Ω24Ω02(|u|η4+|u|η¯4)Ω02C14δ|u|1𝒲12𝒜C32Ω02,less-than-or-similar-tosubscriptnormsuperscript𝑢/ superscriptΩ24superscriptsubscriptΩ02superscript𝑢subscriptnorm𝜂4superscript𝑢subscriptnorm¯𝜂4less-than-or-similar-tosuperscriptsubscriptΩ02superscript𝐶14𝛿superscript𝑢1superscript𝒲12𝒜less-than-or-similar-tosuperscript𝐶32superscriptsubscriptΩ02\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\Omega^{2}\|_{4}\lesssim\Omega_{0}^{2}(|u^{\prime}|\|\eta\|_{4}+|u^{\prime}|\|\underline{\eta}\|_{4})\lesssim\Omega_{0}^{2}C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-\frac{3}{2}}\Omega_{0}^{2}\mathscr{E}, (4.43)

the \engordnumber2 line of (4.41) is estimated by,

less-than-or-similar-to\displaystyle\lesssim δ|u|12Ω01(|u|/ )Ω2𝕃u¯𝕃[u0,u]4Ω0|u|32/ ϕ𝕃u¯𝕃[u0,u]25|u|(|u|/ )Lϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢12subscriptnormsuperscriptsubscriptΩ01𝑢/ superscriptΩ2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnormsubscriptΩ0superscript𝑢32/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-\frac{1}{2}}\|\Omega_{0}^{-1}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\Omega^{2}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\|\Omega_{0}|u|^{\frac{3}{2}}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|12C32Ω0(u0)C14δ12𝒜C14𝒜C1δ22𝒜2(Ω2(u0)δ|u|12)12.less-than-or-similar-to𝛿superscript𝑢12superscript𝐶32subscriptΩ0subscript𝑢0superscript𝐶14superscript𝛿12𝒜superscript𝐶14𝒜superscript𝐶1𝛿superscript2superscript2superscript𝒜2superscriptsuperscriptΩ2subscript𝑢0𝛿superscript𝑢1superscript212\displaystyle\delta|u|^{-\frac{1}{2}}\cdot C^{-\frac{3}{2}}\Omega_{0}(u_{0})\mathscr{E}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\Omega^{2}(u_{0})\delta|u|^{-1}\mathscr{E}^{2})^{\frac{1}{2}}.

The last two terms of the \engordnumber3 line is estimated the same as above, and the \engordnumber1 term is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|2Ω2𝕃u¯𝕃[u0,u]5|u|2η𝕃u¯𝕃[u0,u]4|u|2/ ϕ𝕃u¯𝕃[u0,u]4|u|(|u|/ )Lϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢2subscriptnormsuperscriptΩ2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript5subscriptnormsuperscript𝑢2𝜂subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-2}\|\Omega^{2}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{5}}\||u|^{2}\eta\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|2Ω02(u0)C14δ𝒜C14δ𝒜C14𝒜C1δ22𝒜2(Ω02(u0)δ|u|12)less-than-or-similar-to𝛿superscript𝑢2superscriptsubscriptΩ02subscript𝑢0superscript𝐶14𝛿𝒜superscript𝐶14𝛿𝒜superscript𝐶14𝒜superscript𝐶1𝛿superscript2superscript2superscript𝒜2superscriptsubscriptΩ02subscript𝑢0𝛿superscript𝑢1superscript2\displaystyle\delta|u|^{-2}\cdot\Omega_{0}^{2}(u_{0})\mathscr{E}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\Omega_{0}^{2}(u_{0})\delta|u|^{-1}\mathscr{E}^{2})

The \engordnumber4 line is obtained by taking derivatives on ΩtrχL¯ϕΩtr𝜒¯𝐿italic-ϕ\Omega\mathrm{tr}\chi\underline{L}\phi. It is important that at least one derivative applies on ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi or L¯ϕ¯𝐿italic-ϕ\underline{L}\phi, both of which have worst estimates at zeroth order. The \engordnumber1 term is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|34|u|Ωtrχ𝕃u¯𝕃[u0,u]4Ω0(u0)Ω01|u|34(|u|L¯ϕψ)𝕃u¯𝕃[u0,u]25|u|(|u|/ )Lϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢34subscriptnorm𝑢Ωtr𝜒subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptΩ0subscript𝑢0subscriptnormsuperscriptsubscriptΩ01superscript𝑢34𝑢¯𝐿italic-ϕ𝜓subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-\frac{3}{4}}\||u|\Omega\mathrm{tr}\chi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}(u_{0})\|\Omega_{0}^{-1}|u|^{\frac{3}{4}}(|u|\underline{L}\phi-\psi)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|34𝒜C14δ|u|14Ω0(u0)Ω01𝒜C14𝒜δ22𝒜2(δ|u|1𝒜)14,less-than-or-similar-to𝛿superscript𝑢34𝒜superscript𝐶14𝛿superscript𝑢14subscriptΩ0subscript𝑢0superscriptsubscriptΩ01𝒜superscript𝐶14𝒜𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒜14\displaystyle\delta|u|^{-\frac{3}{4}}\cdot\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-\frac{1}{4}}\Omega_{0}(u_{0})\Omega_{0}^{-1}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{4}},

where we have used both the smallness condition (the second one) and the auxiliary condition (3.5). The \engordnumber2 term is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|2Ω2(trχ2h/|u|)𝕃u¯𝕃[u0,u]5|u|L¯ϕ𝕃u¯𝕃[u0,u]4|u|(|u|/ )Lϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢1subscriptnormsuperscript𝑢2superscriptΩ2trsuperscript𝜒2𝑢subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript5subscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-1}\||u|^{2}\Omega^{2}(\mathrm{tr}\chi^{\prime}-2h/|u|)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{5}}\||u|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C12δ2𝒜2𝒲12C14𝒜𝛿superscript𝑢1superscript𝐶12𝛿superscript2superscript𝒜2superscript𝒲12superscript𝐶14𝒜\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\cdot\mathscr{W}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}
less-than-or-similar-to\displaystyle\lesssim C14δ22𝒜2(δ|u|1𝒲12𝒜)12.superscript𝐶14𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1superscript𝒲12𝒜12\displaystyle C^{-\frac{1}{4}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot\left(\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\right)^{\frac{1}{2}}.
Remark 4.3.

Here we have used the estimate of (|u|/ )5(Ωtrχ)𝕃2(u¯,u)subscriptnormsuperscript𝑢/ 5Ωtr𝜒superscript𝕃2¯𝑢𝑢\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{5}(\Omega\mathrm{tr}\chi)\|_{\mathbb{L}^{2}(\underline{u},u)} which is (the lower order estimates are given in (4.11))

(|u|/ )5(Ωtrχ)𝕃2(u¯,u)(|u|/ )Ω24(u¯,u)trχ4(u¯,u)+Ω02(|u|/ )trχ4(u¯,u)C14δ|u|1𝒲12𝒜|u|1𝒜+C12δ|u|22𝒜2C12δ|u|22𝒲12𝒜2.less-than-or-similar-tosubscriptdelimited-∥∥superscript𝑢/ 5Ωtr𝜒superscript𝕃2¯𝑢𝑢subscriptdelimited-∥∥𝑢/ superscriptΩ2superscript4¯𝑢𝑢subscriptdelimited-∥∥trsuperscript𝜒superscript4¯𝑢𝑢superscriptsubscriptΩ02subscriptdelimited-∥∥𝑢/ trsuperscript𝜒superscript4¯𝑢𝑢less-than-or-similar-tosuperscript𝐶14𝛿superscript𝑢1superscript𝒲12𝒜superscript𝑢1𝒜superscript𝐶12𝛿superscript𝑢2superscript2superscript𝒜2less-than-or-similar-tosuperscript𝐶12𝛿superscript𝑢2superscript2superscript𝒲12superscript𝒜2\begin{split}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{5}(\Omega\mathrm{tr}\chi)\|_{\mathbb{L}^{2}(\underline{u},u)}\lesssim&\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\Omega^{2}\|_{\mathbb{H}^{4}(\underline{u},u)}\|\mathrm{tr}\chi^{\prime}\|_{\mathbb{H}^{4}(\underline{u},u)}+\Omega_{0}^{2}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mathrm{tr}\chi^{\prime}\|_{\mathbb{H}^{4}(\underline{u},u)}\\ \lesssim&C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot|u|^{-1}\mathscr{F}\mathcal{A}+C^{\frac{1}{2}}\delta|u|^{-2}\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\\ \lesssim&C^{\frac{1}{2}}\delta|u|^{-2}\mathscr{F}^{2}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}.\end{split} (4.44)

The last term of (4.41) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|ω𝕃u¯𝕃[u0,u]4Ω0|u|32/ ϕ𝕃u¯𝕃[u0,u]252𝛿superscript𝑢1subscriptnorm𝑢𝜔subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4superscriptsubscriptnormsubscriptΩ0superscript𝑢32/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript52\displaystyle\delta|u|^{-1}\||u|\omega\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\|\Omega_{0}|u|^{\frac{3}{2}}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}}^{2}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C14𝒲12𝒜C12δ22𝒜2𝛿superscript𝑢1superscript𝐶14superscript𝒲12𝒜superscript𝐶12𝛿superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}
less-than-or-similar-to\displaystyle\lesssim C14δ22𝒜2(δ|u|1𝒲12𝒜)12.superscript𝐶14𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1superscript𝒲12𝒜12\displaystyle C^{-\frac{1}{4}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot\left(\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\right)^{\frac{1}{2}}.

Then we turn to the estimates of the right hand side of (4.42) in Lu¯1Lu1superscriptsubscript𝐿¯𝑢1superscriptsubscript𝐿𝑢1L_{\underline{u}}^{1}L_{u}^{1}. The \engordnumber1 term of the \engordnumber1 line is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|2(|u|/ )(Ωχ¯)𝕃u¯𝕃[u0,u]3|u|Lϕ𝕃u¯𝕃[u0,u]4|u|(|u|/ )Lϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢1subscriptnormsuperscript𝑢2𝑢/ Ω¯𝜒subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript3subscriptnorm𝑢𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-1}\||u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\underline{\chi})\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{3}}\||u|L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C14δ𝒜C14𝒜C14𝒜C14δ2𝒜2(δ|u|1𝒜)12.less-than-or-similar-to𝛿superscript𝑢1superscript𝐶14𝛿𝒜superscript𝐶14𝒜superscript𝐶14𝒜superscript𝐶14𝛿superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒜12\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{4}}\delta\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{2}}.

The \engordnumber1 term of the \engordnumber2 line is estimated by (using both (4.44))

less-than-or-similar-to\displaystyle\lesssim δ|u|32Ω0(u0)|u|(|u|/ )(Ωχ)𝕃[u0,u]𝕃u¯24|u|2/ ϕ𝕃u¯𝕃[u0,u]4Ω0|u|32(|u|/ )/ ϕ𝕃u¯𝕃[u0,u]25𝛿superscript𝑢32subscriptΩ0subscript𝑢0subscriptnorm𝑢𝑢/ Ω𝜒subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnormsubscriptΩ0superscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5\displaystyle\delta|u|^{-\frac{3}{2}}\Omega_{0}(u_{0})\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\chi)\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\|\Omega_{0}|u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}}
less-than-or-similar-to\displaystyle\lesssim δ|u|32Ω0(u0)C14𝒜C14δ𝒜C14δ12𝒜C1δ22𝒜2(Ω02(u0)δ|u|12)12.less-than-or-similar-to𝛿superscript𝑢32subscriptΩ0subscript𝑢0superscript𝐶14𝒜superscript𝐶14𝛿𝒜superscript𝐶14superscript𝛿12𝒜superscript𝐶1𝛿superscript2superscript2superscript𝒜2superscriptsuperscriptsubscriptΩ02subscript𝑢0𝛿superscript𝑢1superscript212\displaystyle\delta|u|^{-\frac{3}{2}}\Omega_{0}(u_{0})\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\Omega_{0}^{2}(u_{0})\delta|u|^{-1}\mathscr{E}^{2})^{\frac{1}{2}}.

The estimates of the remaining terms require an estimate of K𝐾K. Similar to (4.35), we can deduce that

Ω0K𝕃u¯24(u¯)|u|2+C38δ|u|332𝒜32|u|212𝒜12.less-than-or-similar-tosubscriptΩ0subscriptnorm𝐾subscriptsuperscript𝕃2¯𝑢superscript4¯𝑢superscript𝑢2superscript𝐶38𝛿superscript𝑢3superscript32superscript𝒜32less-than-or-similar-tosuperscript𝑢2superscript12superscript𝒜12\Omega_{0}\|K\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(\underline{u})}\lesssim|u|^{-2}+C^{\frac{3}{8}}\delta|u|^{-3}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\lesssim|u|^{-2}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}. (4.45)

Then the \engordnumber2 term of the \engordnumber1 line of (4.42) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1Ω0(u0)Ω0|u|2K𝕃[u0,u]𝕃u¯24|u|2/ ϕ𝕃u¯𝕃[u0,u]4|u|(|u|/ )Lϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢1subscriptΩ0subscript𝑢0subscriptnormsubscriptΩ0superscript𝑢2𝐾subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-1}\Omega_{0}(u_{0})\|\Omega_{0}|u|^{2}K\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|112𝒜12C14δ𝒜C14𝒜C12δ22𝒜2(δ|u|1𝒜)12.less-than-or-similar-to𝛿superscript𝑢1superscript12superscript𝒜12superscript𝐶14𝛿𝒜superscript𝐶14𝒜superscript𝐶12𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒜12\displaystyle\delta|u|^{-1}\cdot\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{2}}.

And the last term of (4.42) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|12Ω0(u0)Ω0|u|2K𝕃[u0,u]𝕃u¯23|u|Lϕ𝕃u¯𝕃[u0,u]4|u|32(|u|/ )/ ϕ𝕃u¯𝕃[u0,u]24𝛿superscript𝑢12subscriptΩ0subscript𝑢0subscriptnormsubscriptΩ0superscript𝑢2𝐾subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript3subscriptnorm𝑢𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptnormsuperscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-\frac{1}{2}}\Omega_{0}(u_{0})\|\Omega_{0}|u|^{2}K\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{3}}\||u|L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|12Ω0(u0)12𝒜12C14𝒜C14δ12𝒜δ22𝒜2(δ|u|1𝒜)14.less-than-or-similar-to𝛿superscript𝑢12subscriptΩ0subscript𝑢0superscript12superscript𝒜12superscript𝐶14𝒜superscript𝐶14superscript𝛿12𝒜𝛿superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒜14\displaystyle\delta|u|^{-\frac{1}{2}}\Omega_{0}(u_{0})\cdot\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{4}}.

Now combining all the estimates of (4.41) and (4.42) above, we achieve that (note that / ϕ(u¯=0)=0/ italic-ϕ¯𝑢00\mbox{$\nabla\mkern-13.0mu/$ }\phi(\underline{u}=0)=0)

|u|(|u|/ )Lϕ𝕃u¯24(u)2+δ1|u|32Ω0(|u|/ )/ ϕ𝕃[u0,u]24(u¯)2|u|(|u|/ )Lϕ𝕃u¯24(u0)2+22𝒜2(δ|u|1𝒲(𝒜+Ω02(u0)2))14.less-than-or-similar-tosuperscriptsubscriptdelimited-∥∥𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢2superscript𝛿1superscriptsubscriptdelimited-∥∥superscript𝑢32subscriptΩ0𝑢/ / italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢2superscriptsubscriptdelimited-∥∥𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢02superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒲𝒜superscriptsubscriptΩ02subscript𝑢0superscript214\begin{split}&\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}+\delta^{-1}\||u|^{\frac{3}{2}}\Omega_{0}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}\\ \lesssim&\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2})\right)^{\frac{1}{4}}.\end{split} (4.46)

In particular, this implies

|u|(|u|/ )Lϕ𝕃u¯24(u)+δ12|u|32Ω0(|u|/ )/ ϕ𝕃[u0,u]24(u¯)𝒜.less-than-or-similar-tosubscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢superscript𝛿12subscriptnormsuperscript𝑢32subscriptΩ0𝑢/ / italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢𝒜\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\||u|^{\frac{3}{2}}\Omega_{0}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\mathscr{F}\mathscr{E}\mathcal{A}. (4.47)
Remark 4.4.

The improved estimate (4.46) is crucial in the following parts of the proof.

We consider the last two equations of (4.40). We compute

D¯(|u||(|u|/ )i/ ϕ|2dμg/)+D(Ω2|u||(|u|/ )iL¯ϕ|2dμg/)\displaystyle\underline{D}(|u||(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})+D(\Omega^{-2}|u||(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}\underline{L}\phi|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})
=\displaystyle= |u|2i+1/ A(/ B1BiiL¯ϕ/ i,B1Bi/ Aϕ)+|u|τ2superscript𝑢2𝑖1superscript/ 𝐴subscriptsuperscript/ 𝑖subscript𝐵1subscript𝐵𝑖¯𝐿italic-ϕsuperscript/ 𝑖subscript𝐵1subscript𝐵𝑖subscript/ 𝐴italic-ϕ𝑢subscript𝜏2\displaystyle|u|^{2i+1}\mbox{$\nabla\mkern-13.0mu/$ }^{A}(\mbox{$\nabla\mkern-13.0mu/$ }^{i}_{B_{1}\cdots B_{i}}\underline{L}\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }^{i,B_{1}\cdots B_{i}}\mbox{$\nabla\mkern-13.0mu/$ }_{A}\phi)+|u|\tau_{2}

for 1i51𝑖51\leq i\leq 5, where τ2subscript𝜏2\tau_{2} contains no sixth order derivatives of the derivative of ϕitalic-ϕ\phi. By divergence theorem, we have

δΩ02(u)|u|32(|u|/ )/ ϕ𝕃u¯24(u)2+Ω02(u)|u|2Ω1(|u|/ )L¯ϕ𝕃[u0,u]24(u¯)2𝛿superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢2superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscript𝑢2superscriptΩ1𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢2\displaystyle\delta\Omega_{0}^{2}(u)\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}+\Omega_{0}^{2}(u)\||u|^{2}\Omega^{-1}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}
less-than-or-similar-to\displaystyle\lesssim δΩ02(u)|u|32(|u|/ )/ ϕ𝕃u¯24(u0)2+Ω02(u)|u|2Ω1(|u|/ )L¯ϕ𝕃[u0,u]24(0)2𝛿superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢02superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscript𝑢2superscriptΩ1𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript402\displaystyle\delta\Omega_{0}^{2}(u)\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\Omega_{0}^{2}(u)\||u|^{2}\Omega^{-1}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(0)}^{2}
+0δdu¯u0uduSu¯,uΩ02(u)|u||τ2|dμg/\displaystyle+\int_{0}^{\delta}\mathrm{d}\underline{u}^{\prime}\int_{u_{0}}^{u}\mathrm{d}u^{\prime}\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}||\tau_{2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

where τ2subscript𝜏2\tau_{2} contains no sixth order derivatives of the derivative of ϕitalic-ϕ\phi. By direct computation,

Su¯,uΩ02(u)|u||τ2|dμg/Su¯,uΩ02(u)|u||τ2,1|dμg/+Su¯,uΩ02(u)|u||τ2,2|dμg/\displaystyle\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}||\tau_{2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}||\tau_{2,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}+\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}||\tau_{2,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

where the multiplier terms are

Su¯,uΩ02(u)|u||τ2,1|dμg/Ω02(u)Ω02(u)[|u|3(|u|/ )L¯ϕ4(|Ωtrχ|(|u|/ )L¯ϕ4+(|u|/ )(Ωtrχ)4L¯ϕ4)+|u|3|ω|(|u|/ )L¯ϕ42+|u|3(|u|/ )Ω24(|u|1(|u|/ )/ ϕ4)(|u|/ )L¯ϕ4+|u|3(|u|/ )L¯ϕ4(Ω25η¯4/ ϕ4+Ω24η¯5/ ϕ4+Ω24η¯4/ ϕ5)+|u|3(|u|/ )L¯ϕ4(|Ωtrχ¯|(|u|/ )Lϕ4+(|u|/ )(Ωtrχ¯)4Lϕ4)]+Ω02(u)|u|3(|u|/ )/ ϕ4(|Ωtrχ¯|(|u|/ )/ ϕ4+Ωtrχ¯+2|u|25/ ϕ4)\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}||\tau_{2,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&\Omega_{0}^{2}(u)\Omega_{0}^{-2}(u^{\prime})\left[|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}\left(|\Omega\mathrm{tr}\chi|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}+\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{4}\|\underline{L}\phi\|_{4}\right)\right.\\ &+|u^{\prime}|^{3}|\omega|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}^{2}\\ &+|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\Omega^{2}\|_{4}(|u^{\prime}|^{-1}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4})\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}\\ &+|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}\left(\|\Omega^{2}\|_{5}\|\underline{\eta}\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+\|\Omega^{2}\|_{4}\|\underline{\eta}\|_{5}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+\|\Omega^{2}\|_{4}\|\underline{\eta}\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{5}\right)\\ &\left.+|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}\left(|\Omega\mathrm{tr}\underline{\chi}|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{4}+\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{4}\|L\phi\|_{4}\right)\right]\\ &+\Omega_{0}^{2}(u)|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}(|\Omega\mathrm{tr}\underline{\chi}|\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+\|\Omega\mathrm{tr}\underline{\chi}+2|u^{\prime}|^{-2}\|_{5}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4})\ \end{split} (4.48)

and the commutation terms are

Su¯,u|u||τ2,2|dμg/Ω02(u)Ω02(u)|u|3(|u|/ )L¯ϕ4((|u|/ )(Ωχ)3L¯ϕ4+Ω02(u)|u|K4/ ϕ4)+Ω02(u)|u|3(|u|/ )/ ϕ4((|u|/ )(Ωχ¯)4/ ϕ4+|u|K3(|u|/ )L¯ϕ3).\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u^{\prime}||\tau_{2,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&\Omega_{0}^{2}(u)\Omega_{0}^{-2}(u^{\prime})|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\chi)\|_{3}\|\underline{L}\phi\|_{4}+\Omega_{0}^{2}(u^{\prime})|u^{\prime}|\|K\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\right)\\ &+\Omega_{0}^{2}(u)|u^{\prime}|^{3}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\underline{\chi})\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}+|u^{\prime}|\|K\|_{3}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{3}\right).\end{split} (4.49)

The right hand side of (4.48) and (4.49) should be estimated in Lu¯1Lu1superscriptsubscript𝐿¯𝑢1subscriptsuperscript𝐿1𝑢L_{\underline{u}}^{1}L^{1}_{u}. The \engordnumber1 terms of the first two lines of (4.48) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|Ωtrχ,|u|ω𝕃u¯𝕃[u0,u]4Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]242\displaystyle\delta|u|^{-1}\||u|\Omega\mathrm{tr}\chi,|u|\omega\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|^{2}_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C14𝒲12𝒜C12δ222𝒜2C1δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢1superscript𝐶14superscript𝒲12𝒜superscript𝐶12superscript𝛿2superscript2superscript2superscript𝒜2superscript𝐶1superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{2}}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\lesssim C^{-1}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber2 term of the \engordnumber1 line is estimated by (using (4.44))

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|2Ω2(trχ2h/|u|)𝕃u¯𝕃[u0,u]5|u|L¯ϕ𝕃u¯𝕃[u0,u]24Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24𝛿superscript𝑢1subscriptnormsuperscript𝑢2superscriptΩ2trsuperscript𝜒2𝑢subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript5subscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4subscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-1}\||u|^{2}\Omega^{2}(\mathrm{tr}\chi^{\prime}-2h/|u|)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{5}}\||u|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C12δ2𝒲12𝒜2𝒲12C14δ𝒜C1δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢1superscript𝐶12𝛿superscript2superscript𝒲12superscript𝒜2superscript𝒲12superscript𝐶14𝛿𝒜superscript𝐶1superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}\cdot\mathscr{W}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

In view of (4.43), all terms of the \engordnumber3 and \engordnumber4 lines of (4.48) are estimated in the same manner. They are estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|32Ω0|u|2(/ ϕ,η,η¯)𝕃u¯𝕃[u0,u]4Ω0|u|32(/ ϕ,η¯)𝕃[u0,u]𝕃u¯25Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24𝛿superscript𝑢32subscriptΩ0subscriptnormsuperscript𝑢2/ italic-ϕ𝜂¯𝜂subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptΩ0subscriptnormsuperscript𝑢32/ italic-ϕ¯𝜂subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript5subscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-\frac{3}{2}}\Omega_{0}\||u|^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\phi,\eta,\underline{\eta})\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\||u|^{\frac{3}{2}}(\mbox{$\nabla\mkern-13.0mu/$ }\phi,\underline{\eta})\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|32Ω0C14δ𝒲12𝒜C14δ12𝒲12𝒜C14δ𝒜C1δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢32subscriptΩ0superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14superscript𝛿12superscript𝒲12𝒜superscript𝐶14𝛿𝒜superscript𝐶1superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-\frac{3}{2}}\Omega_{0}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber2 term of the \engordnumber5 line is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1|u|2(Ωtrχ¯+2|u|2)𝕃u¯𝕃[u0,u]5|u|Lϕ𝕃u¯𝕃[u0,u]4Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24𝛿superscript𝑢1subscriptnormsuperscript𝑢2Ωtr¯𝜒2superscript𝑢2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript5subscriptnorm𝑢𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-1}\||u|^{2}(\Omega\mathrm{tr}\underline{\chi}+2|u|^{-2})\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{5}}\||u|L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C14δ𝒲12𝒜C14𝒜C14δ𝒜C1δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢1superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14𝒜superscript𝐶14𝛿𝒜superscript𝐶1superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber2 term of the last line is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|32Ω0|u|2(Ωtrχ¯+2|u|2)𝕃u¯𝕃[u0,u]5|u|2/ ϕ𝕃u¯𝕃[u0,u]4Ω0|u|32(|u|/ )/ ϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢32subscriptΩ0subscriptnormsuperscript𝑢2Ωtr¯𝜒2superscript𝑢2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript5subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptΩ0subscriptnormsuperscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-\frac{3}{2}}\Omega_{0}\||u|^{2}(\Omega\mathrm{tr}\underline{\chi}+2|u|^{-2})\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{5}}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|32Ω0C14δ𝒲12𝒜C14δ𝒜C14δ12𝒜C1δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢32subscriptΩ0superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14𝛿𝒜superscript𝐶14superscript𝛿12𝒜superscript𝐶1superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-\frac{3}{2}}\Omega_{0}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

Now we are going to the crucial terms, the \engordnumber1 terms of the \engordnumber5 line and the last line. Recalling the definition of \mathscr{E}, using (4.46), the \engordnumber1 term of the \engordnumber5 line is estimated by

less-than-or-similar-to\displaystyle\lesssim δ(u0u|u|1||u|Ωtrχ¯|2|u|(|u|/ )Lϕ𝕃u¯24(u)2du)12Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24𝛿superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1superscriptsuperscript𝑢Ωtr¯𝜒2superscriptsubscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4superscript𝑢2differential-dsuperscript𝑢12subscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta\left(\int_{u_{0}}^{u}|u^{\prime}|^{-1}||u^{\prime}|\Omega\mathrm{tr}\underline{\chi}|^{2}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u^{\prime})}^{2}\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ(u0u|u|1(|u|(|u|/ )Lϕ𝕃u¯24(u0)2+22𝒜2(δ|u|1𝒲(𝒜+2))14)du)12𝛿superscriptsuperscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1superscriptsubscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢02superscript2superscript2superscript𝒜2superscript𝛿superscript𝑢1𝒲𝒜superscript214differential-dsuperscript𝑢12\displaystyle\delta\left(\int_{u_{0}}^{u}|u^{\prime}|^{-1}\left(\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\mathscr{E}^{2})\right)^{\frac{1}{4}}\right)\mathrm{d}u^{\prime}\right)^{\frac{1}{2}}
×Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24absentsubscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\times\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ(|log|u1||u0||12|u|(|u|/ )Lϕ𝕃u¯24(u0)+𝒜(δ|u|1𝒲(𝒜+2))18)𝛿superscriptsubscript𝑢1subscript𝑢012subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢0𝒜superscript𝛿superscript𝑢1𝒲𝒜superscript218\displaystyle\delta\left(\left|\log\frac{|u_{1}|}{|u_{0}|}\right|^{\frac{1}{2}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}+\mathscr{F}\mathscr{E}\mathcal{A}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\mathscr{E}^{2})\right)^{\frac{1}{8}}\right)
×Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24absentsubscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\times\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ𝒜Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24.𝛿𝒜subscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}.

Using (4.47), the \engordnumber1 term of the last line is estimated by

δ|u|Ωtrχ𝕃u¯𝕃[u0,u]4Ω0|u|32(|u|/ )/ ϕ𝕃u¯𝕃[u0,u]242δ222𝒜2.less-than-or-similar-toabsent𝛿subscriptnorm𝑢Ωtr𝜒subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptsuperscriptnormsubscriptΩ0superscript𝑢32𝑢/ / italic-ϕ2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4less-than-or-similar-tosuperscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta\||u|\Omega\mathrm{tr}\chi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\|\Omega_{0}|u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|^{2}_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\lesssim\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

Finally, we turn to the estimates of (4.49) in Lu¯1Lu1subscriptsuperscript𝐿1¯𝑢subscriptsuperscript𝐿1𝑢L^{1}_{\underline{u}}L^{1}_{u}. The estimates of the \engordnumber1 terms of both lines are similar to the \engordnumber2 terms of the last two lines of (4.49), and the \engordnumber2 term of the \engordnumber1 line is estimated by (using (4.45))

less-than-or-similar-to\displaystyle\lesssim δ|u|1Ω0Ω0|u|2K𝕃[u0,u]𝕃u¯24|u|2/ ϕ𝕃u¯𝕃[u0,u]4Ω0Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24𝛿superscript𝑢1subscriptΩ0subscriptnormsubscriptΩ0superscript𝑢2𝐾subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptΩ0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\displaystyle\delta|u|^{-1}\Omega_{0}\|\Omega_{0}|u|^{2}K\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1Ω012𝒜12C14δ𝒜C14δ𝒜C1δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢1subscriptΩ0superscript12superscript𝒜12superscript𝐶14𝛿𝒜superscript𝐶14𝛿𝒜superscript𝐶1superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-1}\Omega_{0}\cdot\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

And the last term of (4.49) is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|14Ω0|u|2K𝕃[u0,u]𝕃u¯23|u|74(L¯ϕψ/|u|)𝕃u¯𝕃[u0,u]4Ω0|u|32(|u|/ )/ ϕ𝕃[u0,u]𝕃u¯24𝛿superscript𝑢14subscriptΩ0subscriptnormsuperscript𝑢2𝐾subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript3subscriptnormsuperscript𝑢74¯𝐿italic-ϕ𝜓𝑢subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4subscriptΩ0subscriptnormsuperscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript4\displaystyle\delta|u|^{-\frac{1}{4}}\Omega_{0}\||u|^{2}K\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{3}}\||u|^{\frac{7}{4}}(\underline{L}\phi-\psi/|u|)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}}\cdot\Omega_{0}\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}}
less-than-or-similar-to\displaystyle\lesssim δ|u|1412𝒜12C14δ|u|14𝒜C14δ12𝒜C12δ222𝒜2.less-than-or-similar-to𝛿superscript𝑢14superscript12superscript𝒜12superscript𝐶14𝛿superscript𝑢14𝒜superscript𝐶14superscript𝛿12𝒜superscript𝐶12superscript𝛿2superscript2superscript2superscript𝒜2\displaystyle\delta|u|^{-\frac{1}{4}}\cdot\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\delta|u|^{-\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{2}}\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

Combining all estimates above together, noting that / L¯ϕ(u¯=0)=0/ ¯𝐿italic-ϕ¯𝑢00\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi(\underline{u}=0)=0, we have

δΩ02(u)|u|32(|u|/ )/ ϕ𝕃u¯24(u)2+Ω02(u)|u|2Ω01(|u|/ )L¯ϕ𝕃[u0,u]24(u¯)2δ|u|32(|u|/ )/ ϕ𝕃u¯24(u0)2+δ222𝒜2+δ𝒜Ω0(u)Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24,less-than-or-similar-to𝛿superscriptsubscriptΩ02𝑢superscriptsubscriptdelimited-∥∥superscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢2superscriptsubscriptΩ02𝑢superscriptsubscriptdelimited-∥∥superscript𝑢2superscriptsubscriptΩ01𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢2𝛿superscriptsubscriptdelimited-∥∥superscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢02superscript𝛿2superscript2superscript2superscript𝒜2𝛿𝒜subscriptΩ0𝑢subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4\begin{split}&\delta\Omega_{0}^{2}(u)\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}+\Omega_{0}^{2}(u)\||u|^{2}\Omega_{0}^{-1}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}\\ \lesssim&\delta\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\delta^{2}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}+\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot\Omega_{0}(u)\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}},\end{split}

which implies

Ω0(u)|u|32(|u|/ )/ ϕ𝕃u¯24(u)+δ12Ω0(u)|u|2Ω01(|u|/ )L¯ϕ𝕃[u0,u]24(u¯)δ12𝒜less-than-or-similar-tosubscriptΩ0𝑢subscriptdelimited-∥∥superscript𝑢32𝑢/ / italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢superscript𝛿12subscriptΩ0𝑢subscriptdelimited-∥∥superscript𝑢2superscriptsubscriptΩ01𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢superscript𝛿12𝒜\begin{split}\Omega_{0}(u)\||u|^{\frac{3}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\Omega_{0}(u)\||u|^{2}\Omega_{0}^{-1}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\end{split} (4.50)

The proof of Proposition 4.3 is then complete.  

From Proposition 4.1 and 4.3, we conclude

𝒜.less-than-or-similar-to𝒜\displaystyle\mathcal{E}\lesssim\mathcal{A}.

In particular, ~~\widetilde{\mathcal{E}} in all estimates in Proposition 4.2 can be dropped.

4.6. Estimates for 𝒪~~𝒪\widetilde{\mathcal{O}}

We then use the elliptic systems to estimate 𝒪~~𝒪\widetilde{\mathcal{O}}, which involves the top order derivatives of the connection coefficients.

Proposition 4.4.

Under the assumptions of Theorem 3.1 and the bootstrap assumptions (4.1), we have

𝒪~𝒜+.less-than-or-similar-to~𝒪𝒜\widetilde{\mathcal{O}}\lesssim\mathcal{A}+\mathcal{R}.
Proof.

Estimate for K𝐾K: In order to apply elliptic estimate, we need an appropriate estimate for K𝐾K. Consider the equation for DK𝐷𝐾DK:

D(K|u|2)+ΩtrχK=div/ div/ (Ωχ^)12Δ/ (Ωtrχ).𝐷𝐾superscript𝑢2Ωtr𝜒𝐾div/ div/ Ω^𝜒12Δ/ Ωtr𝜒D(K-|u|^{-2})+\Omega\mathrm{tr}\chi K=\mbox{$\mathrm{div}\mkern-13.0mu/$ }\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})-\frac{1}{2}\mbox{$\Delta\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi). (4.51)

Integrate the equation (4.51), for up to second order derivatives of K𝐾K, using (4.24) and (4.26), we have

K|u|22(u¯,u)less-than-or-similar-tosubscriptnorm𝐾superscript𝑢2superscript2¯𝑢𝑢absent\displaystyle\|K-|u|^{-2}\|_{\mathbb{H}^{2}(\underline{u},u)}\lesssim δ|u|3𝒜+δ2|u|42𝒜2δ|u|3𝒜.less-than-or-similar-to𝛿superscript𝑢3𝒜superscript𝛿2superscript𝑢4superscript2superscript𝒜2𝛿superscript𝑢3𝒜\displaystyle\delta|u|^{-3}\mathscr{F}\mathcal{A}+\delta^{2}|u|^{-4}\mathscr{F}^{2}\mathcal{A}^{2}\lesssim\delta|u|^{-3}\mathscr{F}\mathcal{A}. (4.52)

Using (4.44), we have

K|u|23(u¯,u)C14δ|u|3𝒜+C12δ2|u|4δ2𝒜2C14δ|u|3𝒜.less-than-or-similar-tosubscriptnorm𝐾superscript𝑢2superscript3¯𝑢𝑢superscript𝐶14𝛿superscript𝑢3𝒜superscript𝐶12superscript𝛿2superscript𝑢4𝛿superscript2superscript𝒜2less-than-or-similar-tosuperscript𝐶14𝛿superscript𝑢3𝒜\displaystyle\|K-|u|^{-2}\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim C^{\frac{1}{4}}\delta|u|^{-3}\mathscr{F}\mathscr{E}\mathcal{A}+C^{\frac{1}{2}}\delta^{2}|u|^{-4}\delta\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\lesssim C^{\frac{1}{4}}\delta|u|^{-3}\mathscr{F}\mathscr{E}\mathcal{A}.

This then in particular implies

K2(u¯,u)|u|2,K3(u¯,u)|u|2.formulae-sequenceless-than-or-similar-tosubscriptnorm𝐾superscript2¯𝑢𝑢superscript𝑢2less-than-or-similar-tosubscriptnorm𝐾superscript3¯𝑢𝑢superscript𝑢2\displaystyle\|K\|_{\mathbb{H}^{2}(\underline{u},u)}\lesssim|u|^{-2},\ \|K\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim|u|^{-2}\mathscr{E}. (4.53)
Remark 4.5.

We have a loss \mathscr{E} on the third order derivative of K𝐾K.

Estimates for μ𝜇\mu and η𝜂\eta involving the top order derivatives: Consider first the equation for Dμ𝐷𝜇D\mu,

Dμ+Ωtrχμ=Ωtrχ1|u|2+div/ (2Ωχ^ηΩtrχη¯)+2/ Lϕ/ ϕ+2LϕΔ/ ϕ.𝐷𝜇Ωtr𝜒𝜇Ωtr𝜒1superscript𝑢2div/ 2Ω^𝜒𝜂Ωtr𝜒¯𝜂2/ 𝐿italic-ϕ/ italic-ϕ2𝐿italic-ϕΔ/ italic-ϕD\mu+\Omega\mathrm{tr}\chi\mu=-\Omega\mathrm{tr}\chi\frac{1}{|u|^{2}}+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(2\Omega\widehat{\chi}\cdot\eta-\Omega\mathrm{tr}\chi\underline{\eta})+2\mbox{$\nabla\mkern-13.0mu/$ }L\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi+2L\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi. (4.54)

The right hand side should estimated in δ𝕃u¯14(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}. Noting that Ωχ^,Ωtrχ,LϕΩ^𝜒Ωtr𝜒𝐿italic-ϕ\Omega\widehat{\chi},\Omega\mathrm{tr}\chi,L\phi share a similar bound and η,η¯,/ ϕ𝜂¯𝜂/ italic-ϕ\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi share a similar bound, the last three terms are estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1Ωχ^,Ωtrχ,Lϕ𝕃u¯25(u)η,η¯,/ ϕ𝕃u¯4(u)+δ|u|1χ^,trχ,Lϕ𝕃u¯4(u)η,η¯,/ ϕ𝕃u¯25(u)\displaystyle\delta|u|^{-1}\|\Omega\widehat{\chi},\Omega\mathrm{tr}\chi,L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}\|\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta|u|^{-1}\|\widehat{\chi},\mathrm{tr}\chi,L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}\|\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}
less-than-or-similar-to\displaystyle\lesssim δ|u|1C14|u|1𝒜C14δ|u|2𝒲12𝒜+δ|u|1C14|u|1𝒜C14Ω01δ12|u|32𝒲12𝒜𝛿superscript𝑢1superscript𝐶14superscript𝑢1𝒜superscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜𝛿superscript𝑢1superscript𝐶14superscript𝑢1𝒜superscript𝐶14superscriptsubscriptΩ01superscript𝛿12superscript𝑢32superscript𝒲12𝒜\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}|u|^{-1}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+\delta|u|^{-1}\cdot C^{\frac{1}{4}}|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{3}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}
less-than-or-similar-to\displaystyle\lesssim C12Ω01δ32|u|722𝒲12𝒜2.superscript𝐶12superscriptsubscriptΩ01superscript𝛿32superscript𝑢72superscript2superscript𝒲12superscript𝒜2\displaystyle C^{\frac{1}{2}}\Omega_{0}^{-1}\delta^{\frac{3}{2}}|u|^{-\frac{7}{2}}\mathscr{F}^{2}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}.

Therefore, with |u|2Ωtrχ4(u¯,u)|u|3𝒜less-than-or-similar-tosubscriptnormsuperscript𝑢2Ωtr𝜒superscript4¯𝑢𝑢superscript𝑢3𝒜\||u|^{-2}\Omega\mathrm{tr}\chi\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim|u|^{-3}\mathscr{F}\mathcal{A},

μ4(u¯,u)δ|u|1𝒜μ𝕃u¯4(u¯,u)+Ω01δ|u|332𝒜32.less-than-or-similar-tosubscriptnorm𝜇superscript4¯𝑢𝑢𝛿superscript𝑢1𝒜subscriptnorm𝜇subscriptsuperscript𝕃¯𝑢superscript4¯𝑢𝑢superscriptsubscriptΩ01𝛿superscript𝑢3superscript32superscript𝒜32\displaystyle\|\mu\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\delta|u|^{-1}\mathscr{F}\mathcal{A}\|\mu\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(\underline{u},u)}+\Omega_{0}^{-1}\delta|u|^{-3}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}.

Because δ|u|1𝒜C2C02𝛿superscript𝑢1𝒜superscript𝐶2superscriptsubscript𝐶02\delta|u|^{-1}\mathscr{F}\mathcal{A}\leq C^{-2}\leq C_{0}^{-2}, if C0subscript𝐶0C_{0} is sufficiently large, the first term above can be absorbed by the left hand side, and then we have

μ4(u¯,u)Ω01δ|u|332𝒜32.less-than-or-similar-tosubscriptnorm𝜇superscript4¯𝑢𝑢superscriptsubscriptΩ01𝛿superscript𝑢3superscript32superscript𝒜32\displaystyle\|\mu\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\Omega_{0}^{-1}\delta|u|^{-3}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}. (4.55)

Then from the elliptic system for η𝜂\eta

{div/ η=K1|u|2μcurl/ η=σˇ,casesdiv/ 𝜂𝐾1superscript𝑢2𝜇otherwisecurl/ 𝜂ˇ𝜎otherwise\displaystyle\begin{cases}\mbox{$\mathrm{div}\mkern-13.0mu/$ }\eta=K-\frac{1}{|u|^{2}}-\mu\\ \mbox{$\mathrm{curl}\mkern-13.0mu/$ }\eta=\check{\sigma}\end{cases}, (4.56)

we have, by elliptic estimate and (4.28),

η5(u¯,u)less-than-or-similar-tosubscriptnorm𝜂superscript5¯𝑢𝑢absent\displaystyle\|\eta\|_{\mathbb{H}^{5}(\underline{u},u)}\lesssim |u|K|u|2,σˇ4(u¯,u)+|u|μ4(u¯,u)+η4(u¯,u)\displaystyle|u|\|K-|u|^{-2},\check{\sigma}\|_{\mathbb{H}^{4}(\underline{u},u)}+|u|\|\mu\|_{\mathbb{H}^{4}(\underline{u},u)}+\mathscr{E}\|\eta\|_{\mathbb{H}^{4}(\underline{u},u)}
less-than-or-similar-to\displaystyle\lesssim |u|K|u|2,σˇ4(u¯,u)+Ω01δ|u|2(Ω0(𝒜+)+12𝒜32).\displaystyle|u|\|K-|u|^{-2},\check{\sigma}\|_{\mathbb{H}^{4}(\underline{u},u)}+\Omega_{0}^{-1}\delta|u|^{-2}\mathscr{F}\mathscr{E}(\Omega_{0}\mathscr{E}(\mathcal{A}+\mathcal{R})+\mathscr{F}^{\frac{1}{2}}\mathcal{A}^{\frac{3}{2}}).

Then we have,

η𝕃u¯25(u)|u|K|u|2,σˇ𝕃u¯24(u)+Ω01δ|u|2(C14Ω0𝒜+12𝒜32)C38Ω01δ|u|232𝒜32+δ|u|22(𝒜+)Ω01δ12|u|32(𝒜+),\begin{split}\|\eta\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}\lesssim&|u|\|K-|u|^{-2},\check{\sigma}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\Omega_{0}^{-1}\delta|u|^{-2}\mathscr{F}\mathscr{E}(C^{\frac{1}{4}}\Omega_{0}\mathscr{E}\mathcal{A}+\mathscr{F}^{\frac{1}{2}}\mathcal{A}^{\frac{3}{2}})\\ \lesssim&C^{\frac{3}{8}}\Omega_{0}^{-1}\delta|u|^{-2}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}+\delta|u|^{-2}\mathscr{F}\mathscr{E}^{2}(\mathcal{A}+\mathcal{R})\lesssim\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{3}{2}}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}),\end{split} (4.57)

and

Ω0|u|32η𝕃[u0,u]25(u¯)Ω0|u|52(K|u|2,σˇ)𝕃[u0,u]24(u¯)+δ|u|12(Ω0(𝒜+)+12𝒜32)δ12(𝒜+),less-than-or-similar-tosubscriptdelimited-∥∥subscriptΩ0superscript𝑢32𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢subscriptdelimited-∥∥subscriptΩ0superscript𝑢52𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢𝛿superscript𝑢12subscriptΩ0𝒜superscript12superscript𝒜32less-than-or-similar-tosuperscript𝛿12𝒜\begin{split}\|\Omega_{0}|u|^{\frac{3}{2}}\eta\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}\lesssim&\|\Omega_{0}|u|^{\frac{5}{2}}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}+\delta|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}(\Omega_{0}\mathscr{E}(\mathcal{A}+\mathcal{R})+\mathscr{F}^{\frac{1}{2}}\mathcal{A}^{\frac{3}{2}})\\ \lesssim&\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}),\end{split} (4.58)

Improved estimate for the derivatives of μ𝜇\mu: We can improve the estimates for the derivatives of μ𝜇\mu. This is because the first term |u|2Ωtrχsuperscript𝑢2Ωtr𝜒-|u|^{-2}\Omega\mathrm{tr}\chi has better estimates for its derivatives. We estimate

|u|/ (Ωtrχμ)|u|/ (Ωtrχ)1|u|2𝑢/ Ωtr𝜒𝜇𝑢/ Ωtr𝜒1superscript𝑢2\displaystyle-|u|\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi\mu)-|u|\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi)\frac{1}{|u|^{2}}

in δ𝕃u¯13(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{3}(u)} by, using (4.55),

δ|u|1𝒜(|u|/ )μ𝕃u¯3(u)+δC12δ|u|22𝒜2Ω01|u|212𝒜12.less-than-or-similar-toabsent𝛿superscript𝑢1𝒜subscriptnorm𝑢/ 𝜇subscriptsuperscript𝕃¯𝑢superscript3𝑢𝛿superscript𝐶12𝛿superscript𝑢2superscript2superscript𝒜2superscriptsubscriptΩ01superscript𝑢2superscript12superscript𝒜12\displaystyle\lesssim\delta|u|^{-1}\mathscr{F}\mathcal{A}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mu\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{3}(u)}+\delta\cdot C^{\frac{1}{2}}\delta|u|^{-2}\mathscr{F}^{2}\mathcal{A}^{2}\cdot\Omega_{0}^{-1}|u|^{-2}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}.

Therefore, if C0subscript𝐶0C_{0} is sufficiently large such that the first term can be absorbed, we have, combining with the estimates for the last three terms of (4.54),

(|u|/ )μ3(u¯,u)C12Ω01δ32|u|722𝒲12𝒜2.less-than-or-similar-tosubscriptnorm𝑢/ 𝜇superscript3¯𝑢𝑢superscript𝐶12superscriptsubscriptΩ01superscript𝛿32superscript𝑢72superscript2superscript𝒲12superscript𝒜2\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mu\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim C^{\frac{1}{2}}\Omega_{0}^{-1}\delta^{\frac{3}{2}}|u|^{-\frac{7}{2}}\mathscr{F}^{2}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}. (4.59)

Estimates for μ¯¯𝜇\underline{\mu} and η¯¯𝜂\underline{\eta} involving the top order derivatives: We consider the equation for D¯μ¯¯𝐷¯𝜇\underline{D}\underline{\mu}:

D¯μ¯+Ωtrχ¯μ¯=(Ωtrχ¯+2|u|)1|u|2+div/ (2Ωχ¯^η¯Ωtrχ¯η)+2/ L¯ϕ/ ϕ+2L¯ϕΔ/ ϕ¯𝐷¯𝜇Ωtr¯𝜒¯𝜇Ωtr¯𝜒2𝑢1superscript𝑢2div/ 2Ω^¯𝜒¯𝜂Ωtr¯𝜒𝜂2/ ¯𝐿italic-ϕ/ italic-ϕ2¯𝐿italic-ϕΔ/ italic-ϕ\underline{D}\underline{\mu}+\Omega\mathrm{tr}\underline{\chi}\underline{\mu}=-(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})\frac{1}{|u|^{2}}+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(2\Omega\widehat{\underline{\chi}}\cdot\underline{\eta}-\Omega\mathrm{tr}\underline{\chi}\eta)+2\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi+2\underline{L}\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi (4.60)

We compute the initial value of μ¯¯𝜇\underline{\mu} on Cu0subscript𝐶subscript𝑢0C_{u_{0}} first. Because div/ η¯=K1|u|2μ¯div/ ¯𝜂𝐾1superscript𝑢2¯𝜇\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}=K-\frac{1}{|u|^{2}}-\underline{\mu}, then

μ¯𝕃u¯24(u0)|u|1η¯𝕃u¯25(u0)+K|u|2𝕃u¯24(u0)Ω01(u0)δ12|u|52𝒲12𝒜.less-than-or-similar-tosubscriptnorm¯𝜇subscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢0superscript𝑢1subscriptnorm¯𝜂subscriptsuperscript𝕃2¯𝑢superscript5subscript𝑢0subscriptnorm𝐾superscript𝑢2subscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢0less-than-or-similar-tosuperscriptsubscriptΩ01subscript𝑢0superscript𝛿12superscript𝑢52superscript𝒲12𝒜\displaystyle\|\underline{\mu}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}\lesssim|u|^{-1}\|\underline{\eta}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u_{0})}+\|K-|u|^{-2}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}\lesssim\Omega_{0}^{-1}(u_{0})\delta^{\frac{1}{2}}|u|^{-\frac{5}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (4.61)

So in the following, we estimate μ¯¯𝜇\underline{\mu} in 𝕃u¯24(u)subscriptsuperscript𝕃2¯𝑢superscript4𝑢\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u). The right hand side of (4.60) should be estimated in |u|3𝕃[u0,u]14(u¯)\||u|^{3}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}. The first term is easily estimated by C14δ|u|1𝒜less-than-or-similar-toabsentsuperscript𝐶14𝛿superscript𝑢1𝒜\lesssim C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}. The term div/ (2Ωχ¯^η¯)div/ 2Ω^¯𝜒¯𝜂\mbox{$\mathrm{div}\mkern-13.0mu/$ }(2\Omega\widehat{\underline{\chi}}\cdot\underline{\eta}) is estimated by

|u|32Ω0(u0)Ω01|u|32Ωχ¯^𝕃[u0,u]25(u¯)|u|2η¯𝕃[u0,u]4(u¯)superscript𝑢32subscriptΩ0subscript𝑢0subscriptnormsuperscriptsubscriptΩ01superscript𝑢32Ω^¯𝜒subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢subscriptnormsuperscript𝑢2¯𝜂subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢\displaystyle|u|^{-\frac{3}{2}}\Omega_{0}(u_{0})\|\Omega_{0}^{-1}|u|^{\frac{3}{2}}\Omega\widehat{\underline{\chi}}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}\||u|^{2}\underline{\eta}\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
+Ω01|u|1|u|2Ωχ¯^𝕃[u0,u]4(u¯)Ω0|u|η¯𝕃[u0,u]25(u¯)superscriptsubscriptΩ01superscript𝑢1subscriptnormsuperscript𝑢2Ω^¯𝜒subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢subscriptnormsubscriptΩ0𝑢¯𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢\displaystyle+\Omega_{0}^{-1}|u|^{-1}\||u|^{2}\Omega\widehat{\underline{\chi}}\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\|\Omega_{0}|u|\underline{\eta}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim |u|2Ω0(u0)Ω01C14δ𝒲12𝒜C14δ𝒲12𝒜+Ω01C14|u|1δ𝒜Ω0|u|η¯𝕃[u0,u]25(u¯).superscript𝑢2subscriptΩ0subscript𝑢0superscriptsubscriptΩ01superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14𝛿superscript𝒲12𝒜superscriptsubscriptΩ01superscript𝐶14superscript𝑢1𝛿𝒜delimited-‖|subscriptΩ0𝑢subscriptdelimited-|‖¯𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢\displaystyle|u|^{-2}\Omega_{0}(u_{0})\cdot\Omega_{0}^{-1}C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+\Omega_{0}^{-1}C^{\frac{1}{4}}|u|^{-1}\delta\mathscr{F}\mathcal{A}\|\Omega_{0}|u|\underline{\eta}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}.

The term div/ (Ωtrχ¯η)div/ Ωtr¯𝜒𝜂\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}\eta) is estimated by, using (4.58),

|u|2|u|2(|u|/ )(Ωtrχ¯)𝕃[u0,u]4(u¯)|u|2η¯𝕃[u0,u]4(u¯)superscript𝑢2subscriptnormsuperscript𝑢2𝑢/ Ωtr¯𝜒subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢subscriptnormsuperscript𝑢2¯𝜂subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢\displaystyle|u|^{-2}\||u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\||u|^{2}\underline{\eta}\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
+Ω01|u|12|u|Ωtrχ¯𝕃[u0,u]4(u¯)Ω0|u|32η𝕃[u0,u]25(u¯)superscriptsubscriptΩ01superscript𝑢12subscriptnorm𝑢Ωtr¯𝜒subscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢subscriptnormsubscriptΩ0superscript𝑢32𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢\displaystyle+\Omega_{0}^{-1}|u|^{-\frac{1}{2}}\||u|\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\|\Omega_{0}|u|^{\frac{3}{2}}\eta\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim |u|2Ω01Ω0C14δ𝒲12𝒜C14δ𝒲12𝒜+Ω01|u|121δ12(𝒜+).superscript𝑢2subscriptsuperscriptΩ10subscriptΩ0superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14𝛿superscript𝒲12𝒜superscriptsubscriptΩ01superscript𝑢121superscript𝛿12𝒜\displaystyle|u|^{-2}\Omega^{-1}_{0}\Omega_{0}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+\Omega_{0}^{-1}|u|^{-\frac{1}{2}}\cdot 1\cdot\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}).

The last two terms are estimated by, using (4.47),

|u|2Ω0(u0)Ω01|u|2(|u|/ )L¯ϕ𝕃[u0,u]24(u¯)|u|2/ ϕ𝕃[u0,u]4(u¯)superscript𝑢2subscriptΩ0subscript𝑢0subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢subscriptnormsuperscript𝑢2/ italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢\displaystyle|u|^{-2}\Omega_{0}(u_{0})\|\Omega_{0}^{-1}|u|^{-2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\||u|^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
+Ω01|u|12|u|L¯ϕ𝕃[u0,u]4(u¯)Ω0|u|32/ ϕ𝕃[u0,u]25(u¯)superscriptsubscriptΩ01superscript𝑢12subscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃subscript𝑢0𝑢superscript4¯𝑢subscriptnormsubscriptΩ0superscript𝑢32/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢\displaystyle+\Omega_{0}^{-1}|u|^{-\frac{1}{2}}\||u|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\|\Omega_{0}|u|^{\frac{3}{2}}\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim Ω0(u0)|u|2C14Ω01δ𝒜C14δ𝒜+Ω01|u|12𝒲12δ12𝒜.subscriptΩ0subscript𝑢0superscript𝑢2superscript𝐶14superscriptsubscriptΩ01𝛿𝒜superscript𝐶14𝛿𝒜superscriptsubscriptΩ01superscript𝑢12superscript𝒲12superscript𝛿12𝒜\displaystyle\Omega_{0}(u_{0})|u|^{-2}\cdot C^{\frac{1}{4}}\Omega_{0}^{-1}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}+\Omega_{0}^{-1}|u|^{-\frac{1}{2}}\cdot\mathscr{W}^{\frac{1}{2}}\cdot\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}.

Combining the estimates above, we have, using (4.61),

|u|2μ¯4(u¯,u)less-than-or-similar-tosubscriptnormsuperscript𝑢2¯𝜇superscript4¯𝑢𝑢absent\displaystyle\||u|^{2}\underline{\mu}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim |u|2μ¯4(u¯,u0)+Ω01δ12|u|12𝒲12(𝒜+)+C1Ω01Ω0|u|η¯𝕃[u0,u]25(u¯)subscriptnormsuperscript𝑢2¯𝜇superscript4¯𝑢subscript𝑢0superscriptsubscriptΩ01superscript𝛿12superscript𝑢12superscript𝒲12𝒜superscript𝐶1superscriptsubscriptΩ01subscriptnormsubscriptΩ0𝑢¯𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢\displaystyle\||u|^{2}\underline{\mu}\|_{\mathbb{H}^{4}(\underline{u},u_{0})}+\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R})+C^{-1}\Omega_{0}^{-1}\|\Omega_{0}|u|\underline{\eta}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim Ω01δ12|u|12𝒲12(𝒜+)+C1Ω01Ω0|u|η¯𝕃[u0,u]25(u¯).superscriptsubscriptΩ01superscript𝛿12superscript𝑢12superscript𝒲12𝒜superscript𝐶1superscriptsubscriptΩ01delimited-‖|subscriptΩ0𝑢subscriptdelimited-|‖¯𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢\displaystyle\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R})+C^{-1}\Omega_{0}^{-1}\|\Omega_{0}|u|\underline{\eta}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}.

Therefore, taking 𝕃u¯2\|\cdot\|_{\mathbb{L}^{2}_{\underline{u}}} of the above inequality, we have

|u|2μ¯𝕃u¯24(u)Ω01δ12|u|12𝒲12(𝒜+)+C1Ω01Ω0|u|η¯𝕃[u0,u]2𝕃u¯25Ω01δ12|u|12𝒲12(𝒜+)+C1Ω01C14δ|u|12𝒲12𝒜𝕃[u0,u]2Ω01δ12|u|12𝒲12(𝒜+).less-than-or-similar-tosubscriptdelimited-∥∥superscript𝑢2¯𝜇subscriptsuperscript𝕃2¯𝑢superscript4𝑢superscriptsubscriptΩ01superscript𝛿12superscript𝑢12superscript𝒲12𝒜superscript𝐶1superscriptsubscriptΩ01subscriptdelimited-∥∥subscriptΩ0𝑢¯𝜂subscriptsuperscript𝕃2subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢superscript5less-than-or-similar-tosuperscriptsubscriptΩ01superscript𝛿12superscript𝑢12superscript𝒲12𝒜superscript𝐶1superscriptsubscriptΩ01subscriptdelimited-∥∥superscript𝐶14𝛿superscript𝑢12superscript𝒲12𝒜subscriptsuperscript𝕃2subscript𝑢0𝑢less-than-or-similar-tosuperscriptsubscriptΩ01superscript𝛿12superscript𝑢12superscript𝒲12𝒜\begin{split}\||u|^{2}\underline{\mu}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim&\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R})+C^{-1}\Omega_{0}^{-1}\|\Omega_{0}|u|\underline{\eta}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}}\\ \lesssim&\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R})+C^{-1}\Omega_{0}^{-1}\cdot\|C^{\frac{1}{4}}\delta|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\|_{\mathbb{L}^{2}_{[u_{0},u]}}\\ \lesssim&\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{1}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R}).\end{split} (4.62)

Then from the elliptic system for η¯¯𝜂\underline{\eta}:

{div/ η¯=K1|u|2μ¯curl/ η¯=σˇ,casesdiv/ ¯𝜂𝐾1superscript𝑢2¯𝜇otherwisecurl/ ¯𝜂ˇ𝜎otherwise\begin{cases}\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}=K-\frac{1}{|u|^{2}}-\underline{\mu}\\ \mbox{$\mathrm{curl}\mkern-13.0mu/$ }\underline{\eta}=-\check{\sigma}\end{cases}, (4.63)

we have

η¯5(u¯,u)less-than-or-similar-tosubscriptnorm¯𝜂superscript5¯𝑢𝑢absent\displaystyle\|\underline{\eta}\|_{\mathbb{H}^{5}(\underline{u},u)}\lesssim |u|K|u|2,σˇ4(u¯,u)+|u|μ¯4(u¯,u)+η¯4(u¯,u).\displaystyle|u|\|K-|u|^{-2},\check{\sigma}\|_{\mathbb{H}^{4}(\underline{u},u)}+|u|\|\underline{\mu}\|_{\mathbb{H}^{4}(\underline{u},u)}+\mathscr{E}\|\underline{\eta}\|_{\mathbb{H}^{4}(\underline{u},u)}.

We then have, using (4.30),

η¯𝕃u¯25(u)|u|K|u|2,σˇ𝕃u¯24(u)+|u|μ¯𝕃u¯24(u)+δ|u|22𝒲12(𝒜+𝒜1232)Ω01δ12|u|32𝒲12(𝒜++𝒜1232).\begin{split}\|\underline{\eta}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}\lesssim&|u|\|K-|u|^{-2},\check{\sigma}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+|u|\|\underline{\mu}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta|u|^{-2}\mathscr{F}\mathscr{E}^{2}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})\\ \lesssim&\Omega_{0}^{-1}\delta^{\frac{1}{2}}|u|^{-\frac{3}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}).\end{split} (4.64)

From (4.57), (4.58) and (4.64), the dependence on 𝒪~~𝒪\widetilde{\mathcal{O}} of the estimates (4.24), (4.26), (4.33) and (4.36), and of Proposition 4.2 can now be replaced by +𝒜1232superscript𝒜12superscript32\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}.

Estimates for Ωχ^Ω^𝜒\Omega\widehat{\chi} involving the top order derivatives: Now we consider the equation for div/ (Ωχ^)div/ Ω^𝜒\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi}):

div/ (Ωχ^)=12Ω2/ trχ+Ωχ^η¯+12Ωtrχη(ΩβLϕ/ ϕ).div/ Ω^𝜒12superscriptΩ2/ trsuperscript𝜒Ω^𝜒¯𝜂12Ωtr𝜒𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})=\frac{1}{2}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}+\Omega\widehat{\chi}\cdot\underline{\eta}+\frac{1}{2}\Omega\mathrm{tr}\chi\eta-(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi). (4.65)

The terms on the right hand side are estimated in |u|4(u¯,u)|u|\|\cdot\|_{\mathbb{H}^{4}(\underline{u},u)} by

Ω02|u|/ trχ4(u¯,u)+|u|(Ωχ^η¯4(u¯,u)+Ω02trχη4(u¯,u)+ΩβLϕ/ ϕ4(u¯,u))superscriptsubscriptΩ02subscriptnorm𝑢/ trsuperscript𝜒superscript4¯𝑢𝑢𝑢subscriptnormΩ^𝜒¯𝜂superscript4¯𝑢𝑢subscriptnormsuperscriptsubscriptΩ02trsuperscript𝜒𝜂superscript4¯𝑢𝑢subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle\Omega_{0}^{2}\||u|\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}\|_{\mathbb{H}^{4}(\underline{u},u)}+|u|(\|\Omega\widehat{\chi}\underline{\eta}\|_{\mathbb{H}^{4}(\underline{u},u)}+\|\Omega_{0}^{2}\mathrm{tr}\chi^{\prime}\eta\|_{\mathbb{H}^{4}(\underline{u},u)}+\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)})
less-than-or-similar-to\displaystyle\lesssim C14δ|u|22𝒜2+C14|u|1𝒜C14δ|u|1𝒲12𝒜+ΩβLϕ/ ϕ4(u¯,u)superscript𝐶14𝛿superscript𝑢2superscript2superscript𝒜2superscript𝐶14superscript𝑢1𝒜superscript𝐶14𝛿superscript𝑢1superscript𝒲12𝒜subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}+C^{\frac{1}{4}}|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}
less-than-or-similar-to\displaystyle\lesssim C1|u|1𝒜+|u|ΩβLϕ/ ϕ4(u¯,u).superscript𝐶1superscript𝑢1𝒜𝑢subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle C^{-1}|u|^{-1}\mathscr{F}\mathscr{E}\mathcal{A}+|u|\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}.

Then by elliptic estimate (using (4.24), (4.53)),

Ωχ^5(u¯,u)less-than-or-similar-tosubscriptnormΩ^𝜒superscript5¯𝑢𝑢absent\displaystyle\|\Omega\widehat{\chi}\|_{\mathbb{H}^{5}(\underline{u},u)}\lesssim |u|div/ (Ωχ^)4(u¯,u)+Ωχ^4(u¯,u)𝑢subscriptnormdiv/ Ω^𝜒superscript4¯𝑢𝑢subscriptnormΩ^𝜒superscript4¯𝑢𝑢\displaystyle|u|\|\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})\|_{\mathbb{H}^{4}(\underline{u},u)}+\mathscr{E}\|\Omega\widehat{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}
less-than-or-similar-to\displaystyle\lesssim |u|1(𝒜++𝒜1232)+|u|ΩβLϕ/ ϕ4(u¯,u),superscript𝑢1𝒜superscript𝒜12superscript32𝑢subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle|u|^{-1}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})+|u|\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)},

whose 𝕃u¯2subscriptsuperscript𝕃2¯𝑢\mathbb{L}^{2}_{\underline{u}} norm is then bounded by

Ωχ^𝕃u¯25(u)|u|1(𝒜++𝒜1232).less-than-or-similar-tosubscriptnormΩ^𝜒subscriptsuperscript𝕃2¯𝑢superscript5𝑢superscript𝑢1𝒜superscript𝒜12superscript32\displaystyle\|\Omega\widehat{\chi}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u)}\lesssim|u|^{-1}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}). (4.66)

Estimates for trχtrsuperscript𝜒\mathrm{tr}\chi^{\prime} involving the top order derivatives: Now we consider the equation for D/ trχ𝐷/ trsuperscript𝜒D\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}, written in the following form:

D/ trχ=Ω2(η+η¯)(12(Ωtrχ)2+|Ωχ^|2+2(Lϕ)2)(Ωtrχ)/ trχ2Ω2/ (Ωχ^)Ωχ^4Ω2/ LϕLϕ.𝐷/ trsuperscript𝜒superscriptΩ2𝜂¯𝜂12superscriptΩtr𝜒2superscriptΩ^𝜒22superscript𝐿italic-ϕ2Ωtr𝜒/ trsuperscript𝜒2superscriptΩ2/ Ω^𝜒Ω^𝜒4superscriptΩ2/ 𝐿italic-ϕ𝐿italic-ϕ\begin{split}D\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}=&\Omega^{-2}(\eta+\underline{\eta})\left(\frac{1}{2}(\Omega\mathrm{tr}\chi)^{2}+|\Omega\widehat{\chi}|^{2}+2(L\phi)^{2}\right)\\ &-(\Omega\mathrm{tr}\chi)\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}-2\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\widehat{\chi})\cdot\Omega\widehat{\chi}-4\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }L\phi L\phi.\end{split} (4.67)

which is derived by taking /\nabla\mkern-13.0mu/ to (4.25). Here we use 2/ logΩ=η+η¯2/ Ω𝜂¯𝜂2\mbox{$\nabla\mkern-13.0mu/$ }\log\Omega=\eta+\underline{\eta}. Now consider the fourth order derivative / 4superscript/ 4\mbox{$\nabla\mkern-13.0mu/$ }^{4} of the above equation. To estimate / trχ/ trsuperscript𝜒\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime} in 4superscript4\mathbb{H}^{4}, the right hand side should be estimated in δ𝕃u¯14(u)δ𝕃u¯24(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim\delta\|\cdot\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}. The first line is estimated by

less-than-or-similar-to\displaystyle\lesssim δΩ02η,η¯𝕃u¯4(u)Ωtrχ,Ωχ^,Lϕ𝕃u¯4(u)2\displaystyle\delta\Omega_{0}^{-2}\|\eta,\underline{\eta}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}\|\Omega\mathrm{tr}\chi,\Omega\widehat{\chi},L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}
less-than-or-similar-to\displaystyle\lesssim δΩ02C14δ|u|2𝒲12𝒜C12|u|22𝒜2𝛿superscriptsubscriptΩ02superscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜superscript𝐶12superscript𝑢2superscript2superscript𝒜2\displaystyle\delta\Omega_{0}^{-2}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{2}}|u|^{-2}\mathscr{F}^{2}\mathcal{A}^{2}
less-than-or-similar-to\displaystyle\lesssim C1Ω02δ|u|32𝒜2.superscript𝐶1superscriptsubscriptΩ02𝛿superscript𝑢3superscript2superscript𝒜2\displaystyle C^{-1}\Omega_{0}^{-2}\delta|u|^{-3}\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}.

The \engordnumber2 line involves fifth order derivative of the connection coefficients and Lϕ𝐿italic-ϕL\phi, which should be estimated using (4.16), (4.24), (4.47) and (4.66):

δΩ02Ω02/ trχ𝕃u¯24(u)Ωtrχ𝕃u¯4(u)+δΩ02/ (Ωχ^),/ Lϕ𝕃u¯24(u)Ωχ^,Lϕ𝕃u¯4(u)\displaystyle\delta\Omega_{0}^{-2}\|\Omega_{0}^{2}\mbox{$\nabla\mkern-13.0mu/$ }\mathrm{tr}\chi^{\prime}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\|\Omega\mathrm{tr}\chi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta\Omega_{0}^{-2}\|\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\widehat{\chi}),\mbox{$\nabla\mkern-13.0mu/$ }L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\|\Omega\widehat{\chi},L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}
less-than-or-similar-to\displaystyle\lesssim δΩ02(C14δ|u|32𝒜2|u|1𝒜+|u|2(𝒜++𝒜1232)|u|1(𝒜++𝒜1232)\displaystyle\delta\Omega_{0}^{-2}(C^{\frac{1}{4}}\delta|u|^{-3}\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\cdot|u|^{-1}\mathscr{F}\mathcal{A}+|u|^{-2}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})\cdot|u|^{-1}\mathscr{F}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})
less-than-or-similar-to\displaystyle\lesssim Ω02δ|u|32(𝒜++𝒜1232)2.superscriptsubscriptΩ02𝛿superscript𝑢3superscript2superscript𝒜superscript𝒜12superscript322\displaystyle\Omega_{0}^{-2}\delta|u|^{-3}\mathscr{F}^{2}\mathscr{E}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})^{2}.

Combining the above estimates, we have

(|u|/ )trχ4(u¯,u)Ω02δ|u|22(𝒜++𝒜1232)2.less-than-or-similar-tosubscriptnorm𝑢/ trsuperscript𝜒superscript4¯𝑢𝑢superscriptsubscriptΩ02𝛿superscript𝑢2superscript2superscript𝒜superscript𝒜12superscript322\displaystyle\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mathrm{tr}\chi^{\prime}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\Omega_{0}^{-2}\delta|u|^{-2}\mathscr{F}^{2}\mathscr{E}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})^{2}. (4.68)

Estimates for Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} involving the top order derivatives: The next step is to estimate / 5(Ωtrχ¯)superscript/ 5Ωtr¯𝜒\mbox{$\nabla\mkern-13.0mu/$ }^{5}(\Omega\mathrm{tr}\underline{\chi}). This is the most subtle estimate in the whole argument. Before this, we should first obtain an estimate of / iω¯superscript/ 𝑖¯𝜔\mbox{$\nabla\mkern-13.0mu/$ }^{i}\underline{\omega}. Recall the elliptic-transport system for ω¯¯𝜔\underline{\omega}:

{Δ/ ω¯=ω¯/ +div/ (Ωβ¯+L¯ϕ/ ϕ),Dω¯/ +Ωtrχω¯/ +2Ωχ^/ / ω¯+2div/ (Ωχ^)/ ω¯12div/ (Ωtrχ(Ωβ¯+L¯ϕ/ ϕ))+/ (Ω2)(/ (ρ+16𝐑)+/ σ)+Δ/ (Ω2)(ρ+16𝐑)Δ/ (Ω2(2ηη¯|η|2))div/ (Ωχ^(Ωβ¯+L¯ϕ/ ϕ)2Ωχ¯^(ΩβLϕ/ ϕ)+3Ω2η¯(ρ+16𝐑)3Ω2η¯σ)=div/ {2Ω2/ ϕΔ/ ϕ+L¯ϕ/ Lϕ+Lϕ/ L¯ϕΩtrχ¯Lϕ/ ϕ+2Ωχ¯^/ ϕLϕ+2Ω2η¯/ ϕ/ ϕ+Ω2η¯|/ ϕ|2)}.\displaystyle\begin{cases}\mbox{$\Delta\mkern-13.0mu/$ }\underline{\omega}&=\mbox{$\underline{\omega}\mkern-13.0mu/$ }+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi),\\ D\mbox{$\underline{\omega}\mkern-13.0mu/$ }&+\Omega\mathrm{tr}\chi\mbox{$\underline{\omega}\mkern-13.0mu/$ }+2\Omega\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}+2\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})\cdot\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}-\frac{1}{2}\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi))\\ &+\mbox{$\nabla\mkern-13.0mu/$ }(\Omega^{2})\cdot(\mbox{$\nabla\mkern-13.0mu/$ }(\rho+\frac{1}{6}\mathbf{R})+{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\sigma)+\mbox{$\Delta\mkern-13.0mu/$ }(\Omega^{2})(\rho+\frac{1}{6}\mathbf{R})-\mbox{$\Delta\mkern-13.0mu/$ }(\Omega^{2}(2\eta\cdot\underline{\eta}-|\eta|^{2}))\\ &-\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi}\cdot(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-2\Omega\widehat{\underline{\chi}}\cdot(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+3\Omega^{2}\underline{\eta}(\rho+\frac{1}{6}\mathbf{R})-3\Omega^{2}{}^{*}\underline{\eta}\sigma)\\ \phantom{\Delta}=&-\mbox{$\mathrm{div}\mkern-13.0mu/$ }\{2\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\Delta\mkern-13.0mu/$ }\phi+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }L\phi+L\phi\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi-\Omega\mathrm{tr}\underline{\chi}L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\\ &+2\Omega\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi L\phi+2\Omega^{2}\underline{\eta}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\Omega^{2}\underline{\eta}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2})\}\end{cases}.

Because ω¯¯𝜔\underline{\omega} is a function and satisfies a Poisson equation, the elliptic estimate for / ω¯/ ¯𝜔\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega} does not depend on ω¯¯𝜔\underline{\omega} itself. Therefore no bounds for the lower derivatives of ω¯¯𝜔\underline{\omega} are needed. We have

(|u|/ )ω¯4(u¯,u)|u|2ω¯/ 3(u¯,u)+|u|Ωβ¯+L¯ϕ/ ϕ4(u¯,u).less-than-or-similar-tosubscriptnorm𝑢/ ¯𝜔superscript4¯𝑢𝑢superscript𝑢2subscriptnormω¯/ superscript3¯𝑢𝑢𝑢subscriptnormΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim|u|^{2}\|\mbox{$\underline{\omega}\mkern-13.0mu/$ }\|_{\mathbb{H}^{3}(\underline{u},u)}+|u|\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}. (4.69)

The second equation above looks complicated. We can estimate the right hand side term by term, but it is not hard to see the equation can be written in the following schematic form: (we will write / logΩ|u|1similar-to/ Ωsuperscript𝑢1\mbox{$\nabla\mkern-13.0mu/$ }\log\Omega\sim|u|^{-1}, / 2logΩ/ (η+η¯)similar-tosuperscript/ 2Ω/ 𝜂¯𝜂\mbox{$\nabla\mkern-13.0mu/$ }^{2}\log\Omega\sim\mbox{$\nabla\mkern-13.0mu/$ }(\eta+\underline{\eta}) and use K=14trχtrχ¯+12(χ^,χ¯^)(ρ+16𝐑)+|/ ϕ|2𝐾14tr𝜒tr¯𝜒12^𝜒^¯𝜒𝜌16𝐑superscript/ italic-ϕ2K=-\frac{1}{4}\mathrm{tr}\chi\mathrm{tr}\underline{\chi}+\frac{1}{2}(\widehat{\chi},\widehat{\underline{\chi}})-(\rho+\frac{1}{6}\mathbf{R})+|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}, σˇ=σ12χ^χ¯^ˇ𝜎𝜎12^𝜒^¯𝜒\check{\sigma}=\sigma-\frac{1}{2}\widehat{\chi}\wedge\widehat{\underline{\chi}})

Dω¯/ =Ωtrχ/ 2ω¯+i+j=1/ i(Ωχ^)/ j/ ω¯+i+j=1/ i(Ωχ^orΩtrχ)/ j(Ωβ¯+L¯ϕ/ ϕ)+Ω2i+j=1/ i(ηorη¯)/ j(Korσˇ)+i+j+k=1/ i(ηorη¯)/ j(Ωχ^orΩtrχ)/ k(Ωχ¯^orΩtrχ¯)+Ω2/ (ηorη¯)(ηorη¯)2+Ω2i+j=2/ i(ηorη¯)/ j(ηorη¯)+i+j=1/ i(Ωχ¯^)/ j(ΩβLϕ/ ϕ)+Ω2i+j=1/ i/ ϕ/ j/ 2ϕ+i+j=2/ iL¯ϕ/ jLϕ+Ω2i+j+k=1/ i(ηorη¯)/ j/ ϕ/ k/ ϕ+i+j+k=1/ i(Ωtrχ¯orΩχ¯^)/ jLϕ/ k/ ϕ𝐷ω¯/ Ωtr𝜒superscript/ 2¯𝜔subscript𝑖𝑗1superscript/ 𝑖Ω^𝜒superscript/ 𝑗/ ¯𝜔subscript𝑖𝑗1superscript/ 𝑖Ω^𝜒orΩtr𝜒superscript/ 𝑗Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscriptΩ2subscript𝑖𝑗1superscript/ 𝑖𝜂or¯𝜂superscript/ 𝑗𝐾orˇ𝜎subscript𝑖𝑗𝑘1superscript/ 𝑖𝜂or¯𝜂superscript/ 𝑗Ω^𝜒orΩtr𝜒superscript/ 𝑘Ω^¯𝜒orΩtr¯𝜒superscriptΩ2/ 𝜂or¯𝜂superscript𝜂or¯𝜂2superscriptΩ2subscript𝑖𝑗2superscript/ 𝑖𝜂or¯𝜂superscript/ 𝑗𝜂or¯𝜂subscript𝑖𝑗1superscript/ 𝑖Ω^¯𝜒superscript/ 𝑗Ω𝛽𝐿italic-ϕ/ italic-ϕsuperscriptΩ2subscript𝑖𝑗1superscript/ 𝑖/ italic-ϕsuperscript/ 𝑗superscript/ 2italic-ϕsubscript𝑖𝑗2superscript/ 𝑖¯𝐿italic-ϕsuperscript/ 𝑗𝐿italic-ϕsuperscriptΩ2subscript𝑖𝑗𝑘1superscript/ 𝑖𝜂or¯𝜂superscript/ 𝑗/ italic-ϕsuperscript/ 𝑘/ italic-ϕsubscript𝑖𝑗𝑘1superscript/ 𝑖Ωtr¯𝜒orΩ^¯𝜒superscript/ 𝑗𝐿italic-ϕsuperscript/ 𝑘/ italic-ϕ\begin{split}D\mbox{$\underline{\omega}\mkern-13.0mu/$ }=&\Omega\mathrm{tr}\chi\mbox{$\nabla\mkern-13.0mu/$ }^{2}\underline{\omega}+\sum_{i+j=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\Omega\widehat{\chi})\mbox{$\nabla\mkern-13.0mu/$ }^{j}\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}\\ &+\sum_{i+j=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\Omega\widehat{\chi}\ \text{or}\ \Omega\mathrm{tr}\chi)\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\\ &+\Omega^{2}\sum_{i+j=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\eta\ \text{or}\ \underline{\eta})\mbox{$\nabla\mkern-13.0mu/$ }^{j}(K\ \text{or}\ \check{\sigma})\\ &+\sum_{i+j+k=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\eta\ \text{or}\ \underline{\eta})\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\Omega\widehat{\chi}\ \text{or}\ \Omega\mathrm{tr}\chi)\mbox{$\nabla\mkern-13.0mu/$ }^{k}(\Omega\widehat{\underline{\chi}}\ \text{or}\ \Omega\mathrm{tr}\underline{\chi})+\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }(\eta\ \text{or}\ \underline{\eta})\cdot(\eta\ \text{or}\ \underline{\eta})^{2}\\ &+\Omega^{2}\sum_{i+j=2}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\eta\ \text{or}\ \underline{\eta})\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\eta\ \text{or}\ \underline{\eta})\\ &+\sum_{i+j=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\Omega\widehat{\underline{\chi}})\mbox{$\nabla\mkern-13.0mu/$ }^{j}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\\ &+\Omega^{2}\sum_{i+j=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }^{j}\mbox{$\nabla\mkern-13.0mu/$ }^{2}\phi+\sum_{i+j=2}\mbox{$\nabla\mkern-13.0mu/$ }^{i}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }^{j}L\phi\\ &+\Omega^{2}\sum_{i+j+k=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\eta\ \text{or}\ \underline{\eta})\mbox{$\nabla\mkern-13.0mu/$ }^{j}\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }^{k}\mbox{$\nabla\mkern-13.0mu/$ }\phi+\sum_{i+j+k=1}\mbox{$\nabla\mkern-13.0mu/$ }^{i}(\Omega\mathrm{tr}\underline{\chi}\ \text{or}\ \Omega\widehat{\underline{\chi}})\mbox{$\nabla\mkern-13.0mu/$ }^{j}L\phi\mbox{$\nabla\mkern-13.0mu/$ }^{k}\mbox{$\nabla\mkern-13.0mu/$ }\phi\end{split} (4.70)

We want to estimate ω¯/\underline{\omega}\mkern-13.0mu/ in 3superscript3\mathbb{H}^{3} so the right hand side should be estimated in δ𝕃u¯13(u)δ𝕃u¯23(u)\delta\|\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{3}(u)}\lesssim\delta\|\cdot\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{3}(u)}. The first two lines, placing ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi and Ωχ^Ω^𝜒\Omega\widehat{\chi} in 𝕃u¯4subscriptsuperscript𝕃¯𝑢superscript4\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}, are estimated by

δ|u|2C14|u|1𝒜((|u|/ )ω¯𝕃u¯14(u)+|u|Ωβ¯+L¯ϕ/ ϕ𝕃u¯14(u)).less-than-or-similar-toabsent𝛿superscript𝑢2superscript𝐶14superscript𝑢1𝒜subscriptnorm𝑢/ ¯𝜔subscriptsuperscript𝕃1¯𝑢superscript4𝑢𝑢subscriptnormΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢\displaystyle\lesssim\delta|u|^{-2}\cdot C^{\frac{1}{4}}|u|^{-1}\mathscr{F}\mathcal{A}(\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}+|u|\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}).

We have also used the relation / |u|1similar-to/ superscript𝑢1\mbox{$\nabla\mkern-13.0mu/$ }\sim|u|^{-1} here and will use this in the following steps. The \engordnumber3 line, placing η𝜂\eta or η¯¯𝜂\underline{\eta} in 𝕃u¯4subscriptsuperscript𝕃¯𝑢superscript4\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}, is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1Ω0C14δ|u|2𝒲12𝒜Ω0K,σˇ𝕃u¯24(u)δ|u|1Ω0C14δ|u|2𝒲12𝒜|u|212𝒜12\displaystyle\delta|u|^{-1}\Omega_{0}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot\Omega_{0}\|K,\check{\sigma}\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim\delta|u|^{-1}\Omega_{0}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot|u|^{-2}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}
less-than-or-similar-to\displaystyle\lesssim C12δ|u|4𝒜(Ω02δ|u|12)12.superscript𝐶12𝛿superscript𝑢4𝒜superscriptsuperscriptsubscriptΩ02𝛿superscript𝑢1superscript212\displaystyle C^{-\frac{1}{2}}\delta|u|^{-4}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\Omega_{0}^{2}\delta|u|^{-1}\mathscr{E}^{2})^{\frac{1}{2}}.

The \engordnumber4 line consists of terms with three factors, all of which are placed in 4superscript4\mathbb{H}^{4}. The \engordnumber1 term of this line is estimated by

δ|u|1C14δ|u|2𝒲12𝒜C14|u|1𝒜|u|1C12δ|u|4𝒜(δ|u|1𝒜)12less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜superscript𝐶14superscript𝑢1𝒜superscript𝑢1less-than-or-similar-tosuperscript𝐶12𝛿superscript𝑢4𝒜superscript𝛿superscript𝑢1𝒜12\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}|u|^{-1}\mathscr{F}\mathcal{A}\cdot|u|^{-1}\lesssim C^{-\frac{1}{2}}\delta|u|^{-4}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{2}}

and the \engordnumber2 term of this line is estimated by

Ω02δ|u|1(C14δ|u|2𝒲12𝒜)3C1δ|u|4𝒜(δ|u|1𝒜).less-than-or-similar-toabsentsuperscriptsubscriptΩ02𝛿superscript𝑢1superscriptsuperscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜3less-than-or-similar-tosuperscript𝐶1𝛿superscript𝑢4𝒜𝛿superscript𝑢1𝒜\displaystyle\lesssim\Omega_{0}^{2}\delta|u|^{-1}\cdot(C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A})^{3}\lesssim C^{-1}\delta|u|^{-4}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A}).

Note that / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi shares the same estimate with η,η¯𝜂¯𝜂\eta,\underline{\eta} and Lϕ𝐿italic-ϕL\phi shares the same estimate with Ωχ^Ω^𝜒\Omega\widehat{\chi} for lower order (up to fourth order derivatives), so the last line can be estimated in the same manner as above. The \engordnumber6 line, placing Ωχ¯^Ω^¯𝜒\Omega\widehat{\underline{\chi}} in 𝕃u¯4subscriptsuperscript𝕃¯𝑢superscript4\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}, is estimated by

less-than-or-similar-to\displaystyle\lesssim δ|u|1C14δ|u|2𝒜ΩβLϕ/ ϕ𝕃u¯24(u)δ|u|1C14δ|u|2𝒜C14|u|2𝒜less-than-or-similar-to𝛿superscript𝑢1superscript𝐶14𝛿superscript𝑢2𝒜subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢𝛿superscript𝑢1superscript𝐶14𝛿superscript𝑢2𝒜superscript𝐶14superscript𝑢2𝒜\displaystyle\delta|u|^{-1}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathcal{A}\cdot\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}|u|^{-2}\mathscr{F}\mathscr{E}\mathcal{A}
less-than-or-similar-to\displaystyle\lesssim C12δ|u|4𝒜(δ|u|1𝒜)12.superscript𝐶12𝛿superscript𝑢4𝒜superscript𝛿superscript𝑢1𝒜12\displaystyle C^{-\frac{1}{2}}\delta|u|^{-4}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A})^{\frac{1}{2}}.

It remains to estimate the \engordnumber5 and the \engordnumber7 lines. They contain terms with factors which are second order derivative of η,η¯,/ ϕ,Lϕ,L¯ϕ𝜂¯𝜂/ italic-ϕ𝐿italic-ϕ¯𝐿italic-ϕ\eta,\underline{\eta},\mbox{$\nabla\mkern-13.0mu/$ }\phi,L\phi,\underline{L}\phi, which are of the highest order. The \engordnumber5 line and the \engordnumber1 term of the \engordnumber7 line, placing the highest order η,η¯𝜂¯𝜂\eta,\underline{\eta} and / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi in 𝕃u¯25(u)subscriptsuperscript𝕃2¯𝑢superscript5𝑢\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{5}(u) and the other factor in 𝕃u¯4subscriptsuperscript𝕃¯𝑢superscript4\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}, are estimated by

Ω02δ|u|2C14δ|u|2𝒲12𝒜Ω01C14δ12|u|32𝒲12𝒜C12δ|u|4𝒜(δ|u|1𝒜).less-than-or-similar-tosuperscriptsubscriptΩ02𝛿superscript𝑢2superscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜superscriptsubscriptΩ01superscript𝐶14superscript𝛿12superscript𝑢32superscript𝒲12𝒜superscript𝐶12𝛿superscript𝑢4𝒜𝛿superscript𝑢1𝒜\displaystyle\Omega_{0}^{2}\delta|u|^{-2}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot\Omega_{0}^{-1}C^{\frac{1}{4}}\delta^{\frac{1}{2}}|u|^{-\frac{3}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-\frac{1}{2}}\delta|u|^{-4}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}\mathscr{F}\mathcal{A}).

Finally, we turn the \engordnumber2 term of the \engordnumber7 line, which is the most subtle term. If at least one derivative has applied to L¯ϕ¯𝐿italic-ϕ\underline{L}\phi, then Lϕ𝐿italic-ϕL\phi can be placed in 𝕃u¯4subscriptsuperscript𝕃¯𝑢superscript4\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4} (using (4.16) with ~~\widetilde{\mathcal{E}} dropped) and this term is estimated by

δ|u|2(|u|/ )L¯ϕ𝕃u¯14(u)|u|1𝒜=δ|u|3𝒜(|u|/ )L¯ϕ𝕃u¯14(u).less-than-or-similar-toabsent𝛿superscript𝑢2subscriptnorm𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢superscript𝑢1𝒜𝛿superscript𝑢3𝒜subscriptnorm𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢\displaystyle\lesssim\delta|u|^{-2}\cdot\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\cdot|u|^{-1}\mathscr{F}\mathcal{A}=\delta|u|^{-3}\mathscr{F}\mathcal{A}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}.

If no derivatives have applied to L¯ϕ¯𝐿italic-ϕ\underline{L}\phi, then the term, using (4.46), is estimated by

δ|u|3L¯ϕ𝕃u¯4(u)(|u|(|u|/ )Lϕ𝕃u¯24(u0)+𝒜(δ|u|1𝒲(𝒜+Ω02(u0)2))18).less-than-or-similar-toabsent𝛿superscript𝑢3subscriptnorm¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢superscript4𝑢subscriptnorm𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢0𝒜superscript𝛿superscript𝑢1𝒲𝒜superscriptsubscriptΩ02subscript𝑢0superscript218\displaystyle\lesssim\delta|u|^{-3}\|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}\left(\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}+\mathscr{F}\mathscr{E}\mathcal{A}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2})\right)^{\frac{1}{8}}\right).

Combining all estimates above, and using (4.69), we have

(|u|/ )ω¯4(u¯,u)C14δ|u|1𝒜(|u|/ )ω¯𝕃u¯14(u)+C14δ𝒜Ωβ¯+L¯ϕ/ ϕ𝕃u¯14(u)+|u|Ωβ¯+L¯ϕ/ ϕ4(u¯,u)+δ|u|1𝒜(|u|/ )L¯ϕ𝕃u¯14(u)+δ|u|1L¯ϕ𝕃u¯14(u)(|u|(|u|/ )Lϕ𝕃u¯24(u0)+𝒜(δ|u|1𝒲(𝒜+Ω02(u0)2))18)+δ|u|2𝒜(δ|u|1(𝒜+Ω02(u0)2))12.less-than-or-similar-tosubscriptdelimited-∥∥𝑢/ ¯𝜔superscript4¯𝑢𝑢superscript𝐶14𝛿superscript𝑢1𝒜subscriptdelimited-∥∥𝑢/ ¯𝜔subscriptsuperscript𝕃1¯𝑢superscript4𝑢superscript𝐶14𝛿𝒜subscriptdelimited-∥∥Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢𝑢subscriptdelimited-∥∥Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢𝛿superscript𝑢1𝒜subscriptdelimited-∥∥𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢𝛿superscript𝑢1subscriptdelimited-∥∥¯𝐿italic-ϕsubscriptsuperscript𝕃1¯𝑢superscript4𝑢subscriptdelimited-∥∥𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢0𝒜superscript𝛿superscript𝑢1𝒲𝒜superscriptsubscriptΩ02subscript𝑢0superscript218𝛿superscript𝑢2𝒜superscript𝛿superscript𝑢1𝒜superscriptsubscriptΩ02subscript𝑢0superscript212\begin{split}&\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{H}^{4}(\underline{u},u)}\\ \lesssim&C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}+C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\\ &+|u|\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}+\delta|u|^{-1}\mathscr{F}\mathcal{A}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\\ &+\delta|u|^{-1}\|\underline{L}\phi\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{4}(u)}\left(\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}+\mathscr{F}\mathscr{E}\mathcal{A}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2})\right)^{\frac{1}{8}}\right)\\ &+\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2}))^{\frac{1}{2}}.\end{split} (4.71)

Then we estimate Ω01|u|2𝕃[u0,u]1\|\Omega_{0}^{-1}|u|^{2}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}} of the above quantity, which means that we compute u0uΩ01(u)|u|dusuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ01superscript𝑢superscript𝑢differential-dsuperscript𝑢\int_{u_{0}}^{u}\Omega_{0}^{-1}(u^{\prime})|u^{\prime}|\cdot\mathrm{d}u^{\prime} of the above inequality. Note that 𝕃u¯1subscriptsuperscript𝕃1¯𝑢\mathbb{L}^{1}_{\underline{u}} and 𝕃[u0,u]1subscriptsuperscript𝕃1subscript𝑢0𝑢\mathbb{L}^{1}_{[u_{0},u]} commute and then |u|s𝕃[u0,u]1𝕃u¯1i=|u|s𝕃u¯1𝕃[u0,u]1i|u|s𝕃u¯𝕃[u0,u]2i\||u|^{-s}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{L}^{1}_{\underline{u}}\mathbb{H}^{i}}=\||u|^{-s}\cdot\|_{\mathbb{L}^{1}_{\underline{u}}\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{i}}\lesssim|u|^{-s}\|\cdot\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{i}} if s>0𝑠0s>0, we have

Ω01|u|2(|u|/ )ω¯𝕃[u0,u]14(u¯)C14δ|u|1𝒜Ω01|u|2(|u|/ )ω¯𝕃u¯𝕃[u0,u]14+|u|12Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)𝕃u¯𝕃[u0,u]24+δ|u|1𝒜Ω01|u|2(|u|/ )L¯ϕ𝕃u¯𝕃[u0,u]24+δΩ01(u)1𝕃u¯𝕃[u0,u]24|u|L¯ϕ𝕃u¯𝕃[u0,u]24|u|(|u|/ )Lϕ𝕃u¯24(u0)+δΩ01(u)|u|L¯ϕ𝕃u¯𝕃[u0,u]24𝒜(δ|u|1𝒲(𝒜+Ω02(u0)2))18+δΩ01(u)𝒜(δ|u|1(𝒜+Ω02(u0)2))12.less-than-or-similar-tosubscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝜔subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢superscript𝐶14𝛿superscript𝑢1𝒜subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝜔subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4superscript𝑢12subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢72Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4𝛿superscript𝑢1𝒜subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4𝛿superscriptsubscriptΩ01𝑢subscriptdelimited-∥∥1subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4subscriptdelimited-∥∥𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4subscriptdelimited-∥∥𝑢𝑢/ 𝐿italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢0𝛿superscriptsubscriptΩ01𝑢subscriptdelimited-∥∥𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4𝒜superscript𝛿superscript𝑢1𝒲𝒜superscriptsubscriptΩ02subscript𝑢0superscript218𝛿superscriptsubscriptΩ01𝑢𝒜superscript𝛿superscript𝑢1𝒜superscriptsubscriptΩ02subscript𝑢0superscript212\begin{split}&\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\\ \lesssim&C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}}+|u|^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\\ &+\delta|u|^{-1}\mathscr{F}\mathcal{A}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\\ &+\delta\Omega_{0}^{-1}(u)\|1\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\||u|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\||u|(|u|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}\\ &+\delta\Omega_{0}^{-1}(u)\||u|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2})\right)^{\frac{1}{8}}\\ &+\delta\Omega_{0}^{-1}(u)\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2}))^{\frac{1}{2}}.\end{split} (4.72)

Here 1𝕃u¯𝕃[u0,u]24|log|u1||u0||12less-than-or-similar-tosubscriptnorm1subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4superscriptsubscript𝑢1subscript𝑢012\|1\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\lesssim\left|\log\frac{|u_{1}|}{|u_{0}|}\right|^{\frac{1}{2}} will contribute for the logarithmic loss. We assume

Ω01|u|2(|u|/ )ω¯𝕃u¯𝕃[u0,u]14C14Ω01(u)δ𝒲12𝒜.less-than-or-similar-tosubscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝜔subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4superscript𝐶14superscriptsubscriptΩ01𝑢𝛿superscript𝒲12𝒜\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}}\lesssim C^{\frac{1}{4}}\Omega_{0}^{-1}(u)\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (4.73)

Then (4.72) implies, recalling the definition of \mathscr{E},

Ω01|u|2(|u|/ )ω¯𝕃[u0,u]14(u¯)δ|u|1𝒜C14Ω01(u)δ𝒲12𝒜+|u|12C38Ω01(u)δ3232𝒜32+δ|u|1𝒜Ω01(u)δ𝒜+δΩ01(u)𝒲12𝒜+δΩ01(u)𝒲12𝒜(δ|u|1𝒲(𝒜+Ω02(u0)2))18+δΩ01(u)𝒜(δ|u|1(𝒜+Ω02(u0)2))12δΩ01(u)𝒲12𝒜.less-than-or-similar-tosubscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝜔subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢𝛿superscript𝑢1𝒜superscript𝐶14superscriptsubscriptΩ01𝑢𝛿superscript𝒲12𝒜superscript𝑢12superscript𝐶38superscriptsubscriptΩ01𝑢superscript𝛿32superscript32superscript𝒜32𝛿superscript𝑢1𝒜superscriptsubscriptΩ01𝑢𝛿𝒜𝛿superscriptsubscriptΩ01𝑢superscript𝒲12𝒜𝛿superscriptsubscriptΩ01𝑢superscript𝒲12𝒜superscript𝛿superscript𝑢1𝒲𝒜superscriptsubscriptΩ02subscript𝑢0superscript218𝛿superscriptsubscriptΩ01𝑢𝒜superscript𝛿superscript𝑢1𝒜superscriptsubscriptΩ02subscript𝑢0superscript212less-than-or-similar-to𝛿superscriptsubscriptΩ01𝑢superscript𝒲12𝒜\begin{split}&\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\\ \lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\Omega_{0}^{-1}(u)\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\Omega_{0}^{-1}(u)\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\\ &+\delta|u|^{-1}\mathscr{F}\mathcal{A}\cdot\Omega_{0}^{-1}(u)\delta\mathscr{F}\mathscr{E}\mathcal{A}\\ &+\delta\Omega_{0}^{-1}(u)\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\\ &+\delta\Omega_{0}^{-1}(u)\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot\left(\delta|u|^{-1}\mathscr{W}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2})\right)^{\frac{1}{8}}\\ &+\delta\Omega_{0}^{-1}(u)\mathscr{F}\mathscr{E}\mathcal{A}\cdot(\delta|u|^{-1}(\mathscr{F}\mathcal{A}+\Omega_{0}^{2}(u_{0})\mathscr{E}^{2}))^{\frac{1}{2}}\\ \lesssim&\delta\Omega_{0}^{-1}(u)\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{split} (4.74)

If C0subscript𝐶0C_{0} is sufficiently large, this estimate improves (4.73) and this implies that (4.74) holds without assuming (4.73).

Now we are ready to the estimate for / 5(Ωtrχ¯)superscript/ 5Ωtr¯𝜒\mbox{$\nabla\mkern-13.0mu/$ }^{5}(\Omega\mathrm{tr}\underline{\chi}). Recall the equation for D¯(Ωtrχ¯)¯𝐷Ωtr¯𝜒\underline{D}(\Omega\mathrm{tr}\underline{\chi}):

D¯(Ωtrχ¯)=12(Ωtrχ¯)2+2ω¯Ωtrχ¯|Ωχ¯^|22(L¯ϕ)2.¯𝐷Ωtr¯𝜒12superscriptΩtr¯𝜒22¯𝜔Ωtr¯𝜒superscriptΩ^¯𝜒22superscript¯𝐿italic-ϕ2\underline{D}(\Omega\mathrm{tr}\underline{\chi})=-\frac{1}{2}(\Omega\mathrm{tr}\underline{\chi})^{2}+2\underline{\omega}\Omega\mathrm{tr}\underline{\chi}-|\Omega\widehat{\underline{\chi}}|^{2}-2(\underline{L}\phi)^{2}. (4.75)

Apply /\nabla\mkern-13.0mu/ to the above equation, and write it in the following form:

D¯(Ω2/ (Ωtrχ¯))+Ωtrχ¯(Ω2/ (Ωtrχ¯))=2Ω2/ ω¯(Ωtrχ¯)2Ω2(Ωχ¯^)/ (Ωχ¯^)4Ω2L¯ϕ/ L¯ϕ.¯𝐷superscriptΩ2/ Ωtr¯𝜒Ωtr¯𝜒superscriptΩ2/ Ωtr¯𝜒2superscriptΩ2/ ¯𝜔Ωtr¯𝜒2superscriptΩ2Ω^¯𝜒/ Ω^¯𝜒4superscriptΩ2¯𝐿italic-ϕ/ ¯𝐿italic-ϕ\begin{split}&\underline{D}(\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}))+\Omega\mathrm{tr}\underline{\chi}(\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}))\\ &=2\Omega^{-2}\mbox{$\nabla\mkern-13.0mu/$ }\underline{\omega}(\Omega\mathrm{tr}\underline{\chi})-2\Omega^{-2}(\Omega\widehat{\underline{\chi}})\cdot\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}})-4\Omega^{-2}\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi.\\ \end{split} (4.76)

The right hand side should be estimated in |u|4𝕃[u0,u]14(u¯)\||u|^{4}\cdot\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}. The \engordnumber1 term is estimated by, using Ωtrχ¯4(u¯,u)|u|1less-than-or-similar-tosubscriptnormΩtr¯𝜒superscript4¯𝑢𝑢superscript𝑢1\|\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim|u|^{-1},

Ω2|u|3(|u|/ )ω¯Ωtrχ¯𝕃[u0,u]14(u¯)Ω01Ω01|u|2(|u|/ )ω¯𝕃[u0,u]14(u¯)Ω02δ𝒲12𝒜.less-than-or-similar-tosubscriptnormsuperscriptΩ2superscript𝑢3𝑢/ ¯𝜔Ωtr¯𝜒subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢superscriptsubscriptΩ01subscriptnormsuperscriptsubscriptΩ01superscript𝑢2𝑢/ ¯𝜔subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-tosuperscriptsubscriptΩ02𝛿superscript𝒲12𝒜\displaystyle\|\Omega^{-2}|u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\Omega\mathrm{tr}\underline{\chi}\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\Omega_{0}^{-1}\|\Omega_{0}^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{\omega}\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\Omega_{0}^{-2}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.

The \engordnumber2 term is estimated by,

Ω2|u|3(|u|/ )(Ωχ¯^)(Ωχ¯^)𝕃[u0,u]14(u¯)C14δ𝒜Ω02|u|Ωχ¯^𝕃[u0,u]15(u¯)less-than-or-similar-tosubscriptnormsuperscriptΩ2superscript𝑢3𝑢/ Ω^¯𝜒Ω^¯𝜒subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢superscript𝐶14𝛿𝒜subscriptnormsuperscriptsubscriptΩ02𝑢Ω^¯𝜒subscriptsuperscript𝕃1subscript𝑢0𝑢superscript5¯𝑢\displaystyle\|\Omega^{-2}|u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\underline{\chi}})\cdot(\Omega\widehat{\underline{\chi}})\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\|\Omega_{0}^{-2}|u|\Omega\widehat{\underline{\chi}}\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim Ω01C14δ𝒜|u|12𝕃[u0,u]2Ω01|u|32Ωχ¯^𝕃[u0,u]25(u¯)C12Ω02δ2|u|12𝒲12𝒜2less-than-or-similar-tosuperscriptsubscriptΩ01superscript𝐶14𝛿𝒜subscriptnormsuperscript𝑢12subscriptsuperscript𝕃2subscript𝑢0𝑢subscriptnormsuperscriptsubscriptΩ01superscript𝑢32Ω^¯𝜒subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢superscript𝐶12superscriptsubscriptΩ02superscript𝛿2superscript𝑢1superscript2superscript𝒲12superscript𝒜2\displaystyle\Omega_{0}^{-1}C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\||u|^{-\frac{1}{2}}\|_{\mathbb{L}^{2}_{[u_{0},u]}}\|\Omega_{0}^{-1}|u|^{\frac{3}{2}}\Omega\widehat{\underline{\chi}}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}\lesssim C^{\frac{1}{2}}\Omega_{0}^{-2}\delta^{2}|u|^{-1}\mathscr{F}^{2}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2}
less-than-or-similar-to\displaystyle\lesssim C1Ω02δ𝒲12𝒜.superscript𝐶1superscriptsubscriptΩ02𝛿superscript𝒲12𝒜\displaystyle C^{-1}\Omega_{0}^{-2}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.

The last term is estimated by, using (4.50) and (4.13),

Ω2|u|3(|u|/ )(L¯ϕ)L¯ϕ𝕃[u0,u]14(u¯)subscriptnormsuperscriptΩ2superscript𝑢3𝑢/ ¯𝐿italic-ϕ¯𝐿italic-ϕsubscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢\displaystyle\|\Omega^{-2}|u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\underline{L}\phi)\cdot\underline{L}\phi\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim Ω01Ω1|u|2(|u|/ )(L¯ϕ)𝕃[u0,u]24(u¯)|u|L¯ϕ𝕃[u0,u]24(u¯)superscriptsubscriptΩ01subscriptnormsuperscriptΩ1superscript𝑢2𝑢/ ¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢subscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢\displaystyle\Omega_{0}^{-1}\|\Omega^{-1}|u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\underline{L}\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\||u|\underline{L}\phi\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}
less-than-or-similar-to\displaystyle\lesssim Ω02δ𝒲12𝒜.superscriptsubscriptΩ02𝛿superscript𝒲12𝒜\displaystyle\Omega_{0}^{-2}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.

Combining the above estimates, we have

|u|2(|u|/ )(Ωtrχ¯)4(u¯,u)δ𝒲12𝒜.less-than-or-similar-tosubscriptnormsuperscript𝑢2𝑢/ Ωtr¯𝜒superscript4¯𝑢𝑢𝛿superscript𝒲12𝒜\||u|^{2}(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{\mathbb{H}^{4}(\underline{u},u)}\lesssim\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (4.77)

Estimates for Ωχ¯^Ω^¯𝜒\Omega\widehat{\underline{\chi}} involving the top order derivatives: Consider the equation for div/ (Ωχ¯^)div/ Ω^¯𝜒\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}}):

div/ (Ωχ¯^)=12/ (Ωtrχ¯)+Ωχ¯^η12Ωtrχ¯η+(Ωβ¯+L¯ϕ/ ϕ).div/ Ω^¯𝜒12/ Ωtr¯𝜒Ω^¯𝜒𝜂12Ωtr¯𝜒𝜂Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}})=\frac{1}{2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi})+\Omega\widehat{\underline{\chi}}\cdot\eta-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\eta+(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi). (4.78)

Using (4.77), the right hand side is estimated in 4(u¯,u)superscript4¯𝑢𝑢\mathbb{H}^{4}(\underline{u},u) by, using in particular (4.28),

δ|u|3𝒲12𝒜+|u|1δ|u|3(𝒜+)+Ωβ¯+L¯ϕ/ ϕ4(u¯,u).less-than-or-similar-toabsent𝛿superscript𝑢3superscript𝒲12𝒜superscript𝑢1𝛿superscript𝑢3𝒜subscriptnormΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle\lesssim\delta|u|^{-3}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+|u|^{-1}\cdot\delta|u|^{-3}\mathscr{F}\mathscr{E}(\mathcal{A}+\mathcal{R})+\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}.

By elliptic estimate, using (4.33) by replacing 𝒪~~𝒪\widetilde{\mathcal{O}} by +𝒜1232superscript𝒜12superscript32\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}},

Ωχ¯^5(u¯,u)less-than-or-similar-tosubscriptnormΩ^¯𝜒superscript5¯𝑢𝑢absent\displaystyle\|\Omega\widehat{\underline{\chi}}\|_{\mathbb{H}^{5}(\underline{u},u)}\lesssim |u|div/ (Ωχ¯^)4(u¯,u)+Ωχ¯^4(u¯,u)𝑢subscriptnormdiv/ Ω^¯𝜒superscript4¯𝑢𝑢subscriptnormΩ^¯𝜒superscript4¯𝑢𝑢\displaystyle|u|\|\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}})\|_{\mathbb{H}^{4}(\underline{u},u)}+\mathscr{E}\|\Omega\widehat{\underline{\chi}}\|_{\mathbb{H}^{4}(\underline{u},u)}
less-than-or-similar-to\displaystyle\lesssim δ|u|2𝒲12(𝒜++𝒜1232)+|u|Ωβ¯+L¯ϕ/ ϕ4(u¯,u).𝛿superscript𝑢2superscript𝒲12𝒜superscript𝒜12superscript32𝑢subscriptnormΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscript4¯𝑢𝑢\displaystyle\delta|u|^{-2}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})+|u|\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{4}(\underline{u},u)}.

Then

|u|12Ω01|u|32Ωχ¯^𝕃[u0,u]25(u¯)Ω01δ𝒲12(𝒜++𝒜1232)+|u|12Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]24(u¯)Ω01δ𝒲12(𝒜++𝒜1232)+C38Ω01|u|12δ3232𝒜32Ω01δ𝒲12(𝒜++𝒜1232).less-than-or-similar-tosuperscript𝑢12subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢32Ω^¯𝜒subscriptsuperscript𝕃2subscript𝑢0𝑢superscript5¯𝑢superscriptsubscriptΩ01𝛿superscript𝒲12𝒜superscript𝒜12superscript32superscript𝑢12subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢72Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-tosuperscriptsubscriptΩ01𝛿superscript𝒲12𝒜superscript𝒜12superscript32superscript𝐶38superscriptsubscriptΩ01superscript𝑢12superscript𝛿32superscript32superscript𝒜32less-than-or-similar-tosuperscriptsubscriptΩ01𝛿superscript𝒲12𝒜superscript𝒜12superscript32\begin{split}&|u|^{\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{3}{2}}\Omega\widehat{\underline{\chi}}\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{5}(\underline{u})}\\ \lesssim&\Omega_{0}^{-1}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})+|u|^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{{}_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}}\\ \lesssim&\Omega_{0}^{-1}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}})+C^{\frac{3}{8}}\Omega_{0}^{-1}|u|^{-\frac{1}{2}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\\ \lesssim&\Omega_{0}^{-1}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}(\mathcal{A}+\mathcal{R}+\mathcal{A}^{-\frac{1}{2}}\mathcal{R}^{\frac{3}{2}}).\end{split} (4.79)

This concludes the estimate for / 5(Ωχ¯^)superscript/ 5Ω^¯𝜒\mbox{$\nabla\mkern-13.0mu/$ }^{5}(\Omega\widehat{\underline{\chi}}). We have completed the proof of Proposition 4.4.

 

4.7. Estimates for \mathcal{R}

Finally we turn to the estimates for \mathcal{R}.

Proposition 4.5.

Under the assumptions of Theorem 3.1 and the bootstrap assumptions (4.1), we have

𝒜.less-than-or-similar-to𝒜\mathcal{R}\lesssim\mathcal{A}.
Proof.

The proof is by using the renormalized Bianchi equations. At first, we consider the equations for D¯(ΩβLϕ/ ϕ)¯𝐷Ω𝛽𝐿italic-ϕ/ italic-ϕ\underline{D}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-DK𝐷𝐾DK-Dσˇ𝐷ˇ𝜎D\check{\sigma}.

D(K|u|2)𝐷𝐾superscript𝑢2\displaystyle D(K-|u|^{-2}) +ΩtrχK+div/ (ΩβLϕ/ ϕ)Ωtr𝜒𝐾div/ Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle+\Omega\mathrm{tr}\chi K+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
Ωχ^/ η¯+12Ωtrχdiv/ η¯+(ΩβLϕ/ ϕ)η¯Ωχ^η¯η¯+12Ωtrχ|η¯|2=0Ω^𝜒/ ¯𝜂12Ωtr𝜒div/ ¯𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ¯𝜂Ω^𝜒¯𝜂¯𝜂12Ωtr𝜒superscript¯𝜂20\displaystyle-\Omega\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\underline{\eta}+\frac{1}{2}\Omega\mathrm{tr}\chi\mbox{$\mathrm{div}\mkern-13.0mu/$ }\underline{\eta}+(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\cdot\underline{\eta}-\Omega\widehat{\chi}\cdot\underline{\eta}\cdot\underline{\eta}+\frac{1}{2}\Omega\mathrm{tr}\chi|\underline{\eta}|^{2}=0
Dσˇ𝐷ˇ𝜎\displaystyle D\check{\sigma} +32Ωtrχσˇ+curl/ (ΩβLϕ/ ϕ)+12Ωχ^(η¯^η¯+/ ^η¯)32Ωtr𝜒ˇ𝜎curl/ Ω𝛽𝐿italic-ϕ/ italic-ϕ12Ω^𝜒¯𝜂^tensor-product¯𝜂/ ^tensor-product¯𝜂\displaystyle+\frac{3}{2}\Omega\mathrm{tr}\chi\check{\sigma}+\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\frac{1}{2}\Omega\widehat{\chi}\wedge(\underline{\eta}\widehat{\otimes}\underline{\eta}+\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\underline{\eta})
+η¯(ΩβLϕ/ ϕ)+2/ Lϕ/ ϕ=0¯𝜂Ω𝛽𝐿italic-ϕ/ italic-ϕ2/ 𝐿italic-ϕ/ italic-ϕ0\displaystyle+\underline{\eta}\wedge(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+2\mbox{$\nabla\mkern-13.0mu/$ }L\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi=0
D¯(ΩβLϕ/ ϕ)¯𝐷Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle\underline{D}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) +12Ωtrχ¯(ΩβLϕ/ ϕ)Ωχ¯^(ΩβLϕ/ ϕ)+Ω2/ (K|u|2)Ω2/ σˇ12Ωtr¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕΩ^¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕsuperscriptΩ2/ 𝐾superscript𝑢2superscriptΩ2superscript/ ˇ𝜎\displaystyle+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\Omega\widehat{\underline{\chi}}\cdot(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }(K-|u|^{-2})-\Omega^{2}{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma}
+3Ω2(ηKησˇ)12Ω2(/ (χ^,χ¯^)+/ (χ^χ¯^))32Ω2(η(χ^,χ¯^)+η(χ^χ¯^))3superscriptΩ2𝜂𝐾superscript𝜂ˇ𝜎12superscriptΩ2/ ^𝜒^¯𝜒superscript/ ^𝜒^¯𝜒32superscriptΩ2𝜂^𝜒^¯𝜒superscript𝜂^𝜒^¯𝜒\displaystyle+3\Omega^{2}(\eta K-{}^{*}\eta\check{\sigma})-\frac{1}{2}\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi},\widehat{\underline{\chi}})+{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi}\wedge\widehat{\underline{\chi}}))-\frac{3}{2}\Omega^{2}(\eta(\widehat{\chi},\widehat{\underline{\chi}})+{}^{*}\eta(\widehat{\chi}\wedge\widehat{\underline{\chi}}))
+14Ω2/ (trχtrχ¯)+34Ω2trχtrχ¯η2Ωχ^(Ωβ¯+L¯ϕ/ ϕ)14superscriptΩ2/ tr𝜒tr¯𝜒34superscriptΩ2tr𝜒tr¯𝜒𝜂2Ω^𝜒Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle+\frac{1}{4}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }(\mathrm{tr}\chi\mathrm{tr}\underline{\chi})+\frac{3}{4}\Omega^{2}\mathrm{tr}\chi\mathrm{tr}\underline{\chi}\eta-2\Omega\widehat{\chi}\cdot(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
=\displaystyle= 2Ω2Δ/ ϕ/ ϕ+Ω2/ |/ ϕ|22Ωχ^/ ϕL¯ϕ+ΩtrχL¯ϕ/ ϕ2superscriptΩ2Δ/ italic-ϕ/ italic-ϕsuperscriptΩ2/ superscript/ italic-ϕ22Ω^𝜒/ italic-ϕ¯𝐿italic-ϕΩtr𝜒¯𝐿italic-ϕ/ italic-ϕ\displaystyle-2\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}-2\Omega\widehat{\chi}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\underline{L}\phi+\Omega\mathrm{tr}\chi\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi
2Ω2η/ ϕ/ ϕ+2Ω2η|/ ϕ|2,2superscriptΩ2𝜂/ italic-ϕ/ italic-ϕ2superscriptΩ2𝜂superscript/ italic-ϕ2\displaystyle-2\Omega^{2}\eta\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+2\Omega^{2}\eta|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2},

We compute

D¯(|u|2|(|u|/ )i(ΩβLϕ/ ϕ)|2dμg/)+D(Ω2|u|2|(|(|u|/ )i(K|u|2)|2+|(|u|/ )iσˇ|2)dμg/)\displaystyle\underline{D}(|u|^{2}|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})+D(\Omega^{2}|u|^{2}|(|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2})|^{2}+|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}\check{\sigma}|^{2})\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})
=\displaystyle= |u|2+2i/ A(Ω2/ B1Bii(ΩβLϕ/ ϕ)A/ i,B1Bi(K|u|2)\displaystyle-|u|^{2+2i}\mbox{$\nabla\mkern-13.0mu/$ }^{A}(\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }^{i}_{B_{1}\cdots B_{i}}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)_{A}\mbox{$\nabla\mkern-13.0mu/$ }^{i,B_{1}\cdots B_{i}}(K-|u|^{-2})
+Ω2/ B1Bii(ΩβLϕ/ ϕ)A/ i,B1Biσˇ)+|u|2τ3\displaystyle+\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }^{i}_{B_{1}\cdots B_{i}}{}^{*}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)_{A}\mbox{$\nabla\mkern-13.0mu/$ }^{i,B_{1}\cdots B_{i}}\check{\sigma})+|u|^{2}\tau_{3}

for 0i40𝑖40\leq i\leq 4, where τ3subscript𝜏3\tau_{3} contains no fifth order derivative of curvature components. The weight |u|2superscript𝑢2|u|^{2} respects the coefficient 1212\frac{1}{2} of the term 12Ωtrχ¯(ΩβLϕ/ ϕ)12Ωtr¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕ\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) of the third equation. By divergence theorem, we have

δ|u|2(ΩβLϕ/ ϕ)𝕃u¯24(u)2+|u|52Ω(K|u|2,σˇ)𝕃[u0,u]24(u¯)2𝛿superscriptsubscriptnormsuperscript𝑢2Ω𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢2superscriptsubscriptnormsuperscript𝑢52Ω𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢2\displaystyle\delta\||u|^{2}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}+\||u|^{\frac{5}{2}}\Omega(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}
less-than-or-similar-to\displaystyle\lesssim δ|u|2(ΩβLϕ/ ϕ)𝕃u¯24(u0)2+|u|52Ω(K|u|2,σˇ)𝕃[u0,u]24(0)2𝛿superscriptsubscriptnormsuperscript𝑢2Ω𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢02superscriptsubscriptnormsuperscript𝑢52Ω𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2subscript𝑢0𝑢superscript402\displaystyle\delta\||u|^{2}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\||u|^{\frac{5}{2}}\Omega(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(0)}^{2}
+0δdu¯u0uduSu¯,u|u|2|τ3|dμg/\displaystyle+\int_{0}^{\delta}\mathrm{d}\underline{u}^{\prime}\int_{u_{0}}^{u}\mathrm{d}u^{\prime}\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u|^{2}|\tau_{3}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

where

Su¯,u|u|2|τ3|dμg/Su¯,u|u|2|τ3,1|dμg/+Su¯,u|u|2|τ3,2|dμg/\displaystyle\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u^{\prime}|^{2}|\tau_{3}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u^{\prime}|^{2}|\tau_{3,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}+\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u^{\prime}|^{2}|\tau_{3,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

and the multiplier terms

Su¯,u|u|2|τ3,1|dμg/|u|4ΩβLϕ/ ϕ42Ωtrχ¯+2|u|2,Ωχ¯^4+|u|4Ω02|u|1K|u|2,σˇ4ΩβLϕ/ ϕ4+|u|4Ω02ΩβLϕ/ ϕ4η,η¯4K,K|u|2,σˇ4+|u|4ΩβLϕ/ ϕ4|u|1((|u|/ )(Ωχ^)4Ωχ¯^4+Ωχ^4(|u|/ )(Ωχ¯^)4)+|u|4ΩβLϕ/ ϕ4|u|1((|u|/ )(Ωtrχ)4Ωtrχ¯4+Ωtrχ4(|u|/ )(Ωtrχ¯)4)+|u|4ΩβLϕ/ ϕ4(η4Ωχ^,Ωtrχ4Ωχ¯^,Ωtrχ¯4)+|u|4ΩβLϕ/ ϕ4Ωχ^4Ωβ¯+L¯ϕ/ ϕ4+|u|4Ω02ΩβLϕ/ ϕ4/ ϕ4|u|1/ ϕ5+|u|4ΩβLϕ/ ϕ4(Ωχ^,Ωtrχ4/ ϕ4L¯ϕ4+Ω02η4/ ϕ42)+|u|4Ω02K|u|2,σˇ4Ωχ^,Ωtrχ4(K,K|u|2,σˇ4+|u|1η¯5)+|u|4Ω02K|u|2,σˇ4Ωχ^,Ωtrχ4η¯42+|u|4Ω02σˇ4|u|1Lϕ5/ ϕ4+|u|4Ω02|ω|K|u|2,σˇ42\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u^{\prime}|^{2}|\tau_{3,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}^{2}\|\Omega\mathrm{tr}\underline{\chi}+2|u^{\prime}|^{-2},\Omega\widehat{\underline{\chi}}\|_{4}\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}\cdot|u^{\prime}|^{-1}\|K-|u|^{-2},\check{\sigma}\|_{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\eta,\underline{\eta}\|_{4}\|K,K-|u|^{-2},\check{\sigma}\|_{4}\\ &+|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}|u^{\prime}|^{-1}(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\chi})\|_{4}\|\Omega\widehat{\underline{\chi}}\|_{4}+\|\Omega\widehat{\chi}\|_{4}(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\underline{\chi}})\|_{4})\\ &+|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}|u^{\prime}|^{-1}(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{4}\|\Omega\mathrm{tr}\underline{\chi}\|_{4}+\|\Omega\mathrm{tr}\chi\|_{4}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{4})\\ &+|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|\eta\|_{4}\|\Omega\widehat{\chi},\Omega\mathrm{tr}\chi\|_{4}\|\Omega\widehat{\underline{\chi}},\Omega\mathrm{tr}\underline{\chi}\|_{4}\right)\\ &+|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\Omega\widehat{\chi}\|_{4}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}|u^{\prime}|^{-1}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{5}\\ &+|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|\Omega\widehat{\chi},\Omega\mathrm{tr}\chi\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\underline{L}\phi\|_{4}+\Omega_{0}^{2}\|\eta\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}^{2}\right)\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}\|K-|u|^{-2},\check{\sigma}\|_{4}\|\Omega\widehat{\chi},\Omega\mathrm{tr}\chi\|_{4}(\|K,K-|u|^{-2},\check{\sigma}\|_{4}+|u^{\prime}|^{-1}\|\underline{\eta}\|_{5})\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}\|K-|u|^{-2},\check{\sigma}\|_{4}\|\|\Omega\widehat{\chi},\Omega\mathrm{tr}\chi\|_{4}\|\underline{\eta}\|_{4}^{2}\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}\|\check{\sigma}\|_{4}|u^{\prime}|^{-1}\|L\phi\|_{5}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}|\omega|\|K-|u|^{-2},\check{\sigma}\|_{4}^{2}\end{split} (4.80)

and commutation terms

Su¯,u|u|2|τ3,2|dμg/|u|4ΩβLϕ/ ϕ4((|u|/ )(Ωχ¯)2ΩβLϕ/ ϕ3+Ω02(u)|u|K2K|u|2,σˇ4)+|u|4Ω02(u)K|u|2,σˇ4((|u|/ )(Ωχ)3K|u|2,σˇ3+|u|K3ΩβLϕ/ ϕ4).\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}|u^{\prime}|^{2}|\tau_{3,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&|u^{\prime}|^{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\underline{\chi})\|_{2}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{3}+\Omega_{0}^{2}(u^{\prime})|u^{\prime}|\|K\|_{2}\|K-|u|^{-2},\check{\sigma}\|_{4}\right)\\ &+|u^{\prime}|^{4}\Omega_{0}^{2}(u^{\prime})\|K-|u|^{-2},\check{\sigma}\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\chi)\|_{3}\|K-|u|^{-2},\check{\sigma}\|_{3}+|u^{\prime}|\|K\|_{3}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\right).\end{split} (4.81)

(4.80) and (4.81) should be estimated in Lu¯1Lu1superscriptsubscript𝐿¯𝑢1subscriptsuperscript𝐿1𝑢L_{\underline{u}}^{1}L^{1}_{u}, similar to (4.41), (4.42) and (4.48) and (4.49), although more terms are needed to estimate. To do the estimates, the basic rule is that, we always place both |u|2ΩβLϕ/ ϕ4superscriptsuperscript𝑢2subscriptnormΩ𝛽𝐿italic-ϕ/ italic-ϕ4|u^{\prime}|^{2}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4} and Ω0|u|3K|u|2,σˇ4\Omega_{0}|u^{\prime}|^{3}\|K-|u|^{-2},\check{\sigma}\|_{4} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢superscriptsubscript𝕃¯𝑢2\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}_{\underline{u}}^{2}. The lower order derivatives of the connection coefficients and Lϕ𝐿italic-ϕL\phi, / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi should be placed in 𝕃u¯𝕃[u0,u]4superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢superscript4\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{H}^{4}.

The \engordnumber1 line of the right hand side is estimated by,

δ|u|1C1222𝒜2C14δ𝒜C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶12superscript2superscript2superscript𝒜2superscript𝐶14𝛿𝒜less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{2}}\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber2 line is estimated by,

δ|u|1C38δ32𝒜32C14𝒜C1δ2𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶38𝛿superscript32superscript𝒜32superscript𝐶14𝒜less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}.

The \engordnumber3 line is estimated by, using (4.45) to estimate K𝐾K,

Ω0(u0)δ|u|1C14𝒜C14δ𝒲12𝒜12𝒜12C12δ22𝒜2.less-than-or-similar-toabsentsubscriptΩ0subscript𝑢0𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝛿superscript𝒲12𝒜superscript12superscript𝒜12less-than-or-similar-tosuperscript𝐶12𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\Omega_{0}(u_{0})\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\lesssim C^{-\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber1 term of the \engordnumber4 line is estimated by, placing |u|(|u|/ )(Ωχ^)4superscript𝑢subscriptnormsuperscript𝑢/ Ω^𝜒4|u^{\prime}|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\chi})\|_{4} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢superscriptsubscript𝕃¯𝑢2\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}_{\underline{u}}^{2},

δ|u|1C14𝒜C14𝒜C14δ𝒜C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝒜superscript𝐶14𝛿𝒜less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber1 term of the \engordnumber5 line is estimated by, placing |u|(|u|/ )(Ωtrχ)5superscript𝑢subscriptnormsuperscript𝑢/ Ωtr𝜒5|u^{\prime}|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{5} in 𝕃u¯𝕃[u0,u]superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{\infty}_{[u_{0},u]}

δ|u|1C14𝒜C12δ2𝒜21C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶12𝛿superscript2superscript𝒜21less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\cdot 1\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber2 terms of the \engordnumber4 and \engordnumber5 lines are estimated by, placing Ω01|u|32(|u|/ )(Ωχ¯^)5subscriptnormsuperscriptsubscriptΩ01superscriptsuperscript𝑢32superscript𝑢/ Ω^¯𝜒5\|\Omega_{0}^{-1}|u^{\prime}|^{\frac{3}{2}}(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\underline{\chi}})\|_{5} in 𝕃u¯𝕃[u0,u]2superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{2}_{[u_{0},u]}, and |u|2(|u|/ )(Ωtrχ¯)5subscriptnormsuperscriptsuperscript𝑢2superscript𝑢/ Ωtr¯𝜒5\||u^{\prime}|^{2}(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{5} in 𝕃u¯𝕃[u0,u]superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{\infty}_{[u_{0},u]}, and using the auxiliary condition (3.5),

δ|u|1C14𝒜C14𝒜C14Ω0(u0)Ω01δ𝒲12𝒜C14δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝒜superscript𝐶14subscriptΩ0subscript𝑢0superscriptsubscriptΩ01𝛿superscript𝒲12𝒜less-than-or-similar-tosuperscript𝐶14𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\Omega_{0}(u_{0})\Omega_{0}^{-1}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-\frac{1}{4}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber6 line is estimated by

δ|u|1C14𝒜C14δ𝒜C14𝒜1C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝛿𝒜superscript𝐶14𝒜1less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot 1\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber7 line is estimated by, placing Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)4subscriptnormsuperscriptsubscriptΩ01superscript𝑢72Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ4\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{4} in 𝕃u¯𝕃[u0,u]2superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{2}_{[u_{0},u]},

δ|u|32C14𝒜C14𝒜C14Ω0(u0)Ω01δ3232𝒜32C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢32superscript𝐶14𝒜superscript𝐶14𝒜superscript𝐶14subscriptΩ0subscript𝑢0superscriptsubscriptΩ01superscript𝛿32superscript32superscript𝒜32less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\Omega_{0}(u_{0})\Omega_{0}^{-1}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

Here we have used the auxiliary condition (3.5). The \engordnumber8 line is estimated by, placing Ω0|u|32/ ϕ5subscriptΩ0superscript𝑢32subscriptnorm/ italic-ϕ5\Omega_{0}|u|^{\frac{3}{2}}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{5} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢superscriptsubscript𝕃¯𝑢2\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}_{\underline{u}}^{2},

Ω0(u0)δ|u|32C14𝒜C14δ𝒜C14δ12𝒜C1δ22𝒜2.less-than-or-similar-toabsentsubscriptΩ0subscript𝑢0𝛿superscript𝑢32superscript𝐶14𝒜superscript𝐶14𝛿𝒜superscript𝐶14superscript𝛿12𝒜less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\Omega_{0}(u_{0})\delta|u|^{-\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber1 term of the \engordnumber9 line is estimated by, using |u|L¯ϕ𝕃u¯𝕃[u0,u]24𝒲12less-than-or-similar-tosubscriptnorm𝑢¯𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4superscript𝒲12\||u|\underline{L}\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}}\lesssim\mathscr{W}^{\frac{1}{2}},

δ|u|1C14𝒜C14𝒜C14δ𝒜𝒲12C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶14𝒜superscript𝐶14𝒜superscript𝐶14𝛿𝒜superscript𝒲12less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot\mathscr{W}^{\frac{1}{2}}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber2 term of the \engordnumber9 line is estimated by,

δ|u|3C14𝒜(C14δ𝒜)3C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢3superscript𝐶14𝒜superscriptsuperscript𝐶14𝛿𝒜3less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-3}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot(C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A})^{3}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber10 and \engordnumber13 lines are estimated in the same manner. ω𝜔\omega has the worst estimate among ω,Ωχ^,Ωtrχ𝜔Ω^𝜒Ωtr𝜒\omega,\Omega\widehat{\chi},\Omega\mathrm{tr}\chi, and Ω0KsubscriptΩ0𝐾\Omega_{0}K has the worst estimate among Ω0K,Ω0(K|u|2),Ω0σˇ,Ω0|u|1η¯subscriptΩ0𝐾subscriptΩ0𝐾superscript𝑢2subscriptΩ0ˇ𝜎subscriptΩ0superscript𝑢1¯𝜂\Omega_{0}K,\Omega_{0}(K-|u|^{-2}),\Omega_{0}\check{\sigma},\Omega_{0}|u|^{-1}\underline{\eta} where Ω0|u|1η¯5subscriptΩ0superscript𝑢1subscriptnorm¯𝜂5\Omega_{0}|u|^{-1}\|\underline{\eta}\|_{5} is placed in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}}. Then these two lines are estimated by

δ|u|1C38δ32𝒜32C14𝒲12𝒜12𝒜12C1δ22𝒜2.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶38𝛿superscript32superscript𝒜32superscript𝐶14superscript𝒲12𝒜superscript12superscript𝒜12less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber11 line is estimated by

Ω0(u0)δ|u|3C38δ32𝒜32C14𝒜(C14δ𝒜)2C1δ22𝒜2.less-than-or-similar-toabsentsubscriptΩ0subscript𝑢0𝛿superscript𝑢3superscript𝐶38𝛿superscript32superscript𝒜32superscript𝐶14𝒜superscriptsuperscript𝐶14𝛿𝒜2less-than-or-similar-tosuperscript𝐶1𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\Omega_{0}(u_{0})\delta|u|^{-3}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot(C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A})^{2}\lesssim C^{-1}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

The \engordnumber12 line is estimated by, placing |u|Lϕ5𝑢subscriptnorm𝐿italic-ϕ5|u|\|L\phi\|_{5} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}},

Ω0(u0)δ|u|2C38δ32𝒜32C14𝒜C14δ𝒜C18δ22𝒜2.less-than-or-similar-toabsentsubscriptΩ0subscript𝑢0𝛿superscript𝑢2superscript𝐶38𝛿superscript32superscript𝒜32superscript𝐶14𝒜superscript𝐶14𝛿𝒜less-than-or-similar-tosuperscript𝐶18𝛿superscript2superscript2superscript𝒜2\displaystyle\lesssim\Omega_{0}(u_{0})\delta|u|^{-2}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}.

Using the estimates (4.53) for K𝐾K in 3(u¯,u)superscript3¯𝑢𝑢\mathbb{H}^{3}(\underline{u},u), (4.81) can be estimated in a similar way, by δ22𝒜2less-than-or-similar-toabsent𝛿superscript2superscript2superscript𝒜2\lesssim\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2}. Then we have, for K(u¯=0)=|u|2𝐾¯𝑢0superscript𝑢2K(\underline{u}=0)=|u|^{-2},

δ|u|2(ΩβLϕ/ ϕ)𝕃u¯24(u)2+|u|52Ω0(K|u|2,σˇ)𝕃[u0,u]24(u¯)2δ|u|2(ΩβLϕ/ ϕ)𝕃u¯24(u0)2+δ22𝒜2,less-than-or-similar-to𝛿superscriptsubscriptdelimited-∥∥superscript𝑢2Ω𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢2superscriptsubscriptdelimited-∥∥superscript𝑢52subscriptΩ0𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢2𝛿superscriptsubscriptdelimited-∥∥superscript𝑢2Ω𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4subscript𝑢02𝛿superscript2superscript2superscript𝒜2\begin{split}&\delta\||u|^{2}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}^{2}+\||u|^{\frac{5}{2}}\Omega_{0}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}^{2}\\ \lesssim&\delta\||u|^{2}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u_{0})}^{2}+\delta\mathscr{F}^{2}\mathscr{E}^{2}\mathcal{A}^{2},\end{split}

that is,

|u|2(ΩβLϕ/ ϕ)𝕃u¯24(u)+δ12|u|52Ω0(K|u|2,σˇ)𝕃[u0,u]24(u¯)𝒜.less-than-or-similar-tosubscriptdelimited-∥∥superscript𝑢2Ω𝛽𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2¯𝑢superscript4𝑢superscript𝛿12subscriptdelimited-∥∥superscript𝑢52subscriptΩ0𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2subscript𝑢0𝑢superscript4¯𝑢𝒜\begin{split}&\||u|^{2}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(u)}+\delta^{-\frac{1}{2}}\||u|^{\frac{5}{2}}\Omega_{0}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\lesssim\mathscr{F}\mathscr{E}\mathcal{A}.\end{split} (4.82)

Finally, we consider the equations for D¯K¯𝐷𝐾\underline{D}K-D¯σˇ¯𝐷ˇ𝜎\underline{D}\check{\sigma}-D(Ωβ¯+L¯ϕ/ ϕ)𝐷Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕD(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi):

D¯(K1|u|2)¯𝐷𝐾1superscript𝑢2\displaystyle\underline{D}(K-\frac{1}{|u|^{2}}) +32Ωtrχ¯(K1|u|2)+(Ωtrχ¯+2|u|)1|u|232Ωtr¯𝜒𝐾1superscript𝑢2Ωtr¯𝜒2𝑢1superscript𝑢2\displaystyle+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}(K-\frac{1}{|u|^{2}})+(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})\frac{1}{|u|^{2}}
=\displaystyle= div/ (Ωβ¯+L¯ϕ/ ϕ)+Ωχ¯^/ η+12Ωtrχ¯μdiv/ Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕΩ^¯𝜒/ 𝜂12Ωtr¯𝜒𝜇\displaystyle\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)+\Omega\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\eta+\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\mu
+(Ωβ¯+L¯ϕ/ ϕ)η+Ωχ¯^ηη12Ωtrχ¯|η|2,Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ𝜂Ω^¯𝜒𝜂𝜂12Ωtr¯𝜒superscript𝜂2\displaystyle+(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\cdot\eta+\Omega\widehat{\underline{\chi}}\cdot\eta\cdot\eta-\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}|\eta|^{2},
D¯σˇ¯𝐷ˇ𝜎\displaystyle\underline{D}\check{\sigma} +32Ωtrχσˇ+curl/ (Ωβ¯+L¯ϕ/ ϕ)32Ωtr𝜒ˇ𝜎curl/ Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle+\frac{3}{2}\Omega\mathrm{tr}\chi\check{\sigma}+\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
12Ωχ¯^(η^η+/ ^η)+η(Ωβ¯+L¯ϕ/ ϕ)2/ L¯ϕ/ ϕ=012Ω^¯𝜒𝜂^tensor-product𝜂/ ^tensor-product𝜂𝜂Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ2/ ¯𝐿italic-ϕ/ italic-ϕ0\displaystyle-\frac{1}{2}\Omega\widehat{\underline{\chi}}\wedge(\eta\widehat{\otimes}\eta+\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\eta)+\eta\wedge(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-2\mbox{$\nabla\mkern-13.0mu/$ }\underline{L}\phi\wedge\mbox{$\nabla\mkern-13.0mu/$ }\phi=0
D(Ωβ¯+L¯ϕ/ ϕ)𝐷Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\displaystyle D(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) +12Ωtrχ(Ωβ¯+L¯ϕ/ ϕ)Ωχ^(Ωβ¯+L¯ϕ/ ϕ)Ω2/ KΩ2/ σˇ12Ωtr𝜒Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕΩ^𝜒Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsuperscriptΩ2/ 𝐾superscriptΩ2superscript/ ˇ𝜎\displaystyle+\frac{1}{2}\Omega\mathrm{tr}\chi(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\Omega\widehat{\chi}\cdot(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }K-\Omega^{2}{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma}
3Ω2(η¯K+η¯σˇ)+12Ω2(/ (χ^,χ¯^)/ (χ^χ¯^))+32Ω2(η¯(χ^,χ¯^)+η¯(χ^χ¯^))3superscriptΩ2¯𝜂𝐾superscript¯𝜂ˇ𝜎12superscriptΩ2/ ^𝜒^¯𝜒superscript/ ^𝜒^¯𝜒32superscriptΩ2¯𝜂^𝜒^¯𝜒superscript¯𝜂^𝜒^¯𝜒\displaystyle-3\Omega^{2}(\underline{\eta}K+{}^{*}\underline{\eta}\check{\sigma})+\frac{1}{2}\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi},\widehat{\underline{\chi}})-{}^{*}\mbox{$\nabla\mkern-13.0mu/$ }(\widehat{\chi}\wedge\widehat{\underline{\chi}}))+\frac{3}{2}\Omega^{2}(\underline{\eta}(\widehat{\chi},\widehat{\underline{\chi}})+{}^{*}\underline{\eta}(\widehat{\chi}\wedge\widehat{\underline{\chi}}))
14Ω2/ (trχtrχ¯)34Ω2trχtrχ¯η¯2Ωχ¯^(ΩβLϕ/ ϕ)14superscriptΩ2/ tr𝜒tr¯𝜒34superscriptΩ2tr𝜒tr¯𝜒¯𝜂2Ω^¯𝜒Ω𝛽𝐿italic-ϕ/ italic-ϕ\displaystyle-\frac{1}{4}\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }(\mathrm{tr}\chi\mathrm{tr}\underline{\chi})-\frac{3}{4}\Omega^{2}\mathrm{tr}\chi\mathrm{tr}\underline{\chi}\underline{\eta}-2\Omega\widehat{\underline{\chi}}\cdot(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)
=\displaystyle= 2Ω2Δ/ ϕ/ ϕΩ2/ |/ ϕ|2+2Ωχ¯^/ ϕLϕΩtrχ¯Lϕ/ ϕ2superscriptΩ2Δ/ italic-ϕ/ italic-ϕsuperscriptΩ2/ superscript/ italic-ϕ22Ω^¯𝜒/ italic-ϕ𝐿italic-ϕΩtr¯𝜒𝐿italic-ϕ/ italic-ϕ\displaystyle 2\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi-\Omega^{2}\mbox{$\nabla\mkern-13.0mu/$ }|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}+2\Omega\widehat{\underline{\chi}}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi L\phi-\Omega\mathrm{tr}\underline{\chi}L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi
+2Ω2η¯/ ϕ/ ϕ2Ω2η¯|/ ϕ|2.2superscriptΩ2¯𝜂/ italic-ϕ/ italic-ϕ2superscriptΩ2¯𝜂superscript/ italic-ϕ2\displaystyle+2\Omega^{2}\underline{\eta}\cdot\mbox{$\nabla\mkern-13.0mu/$ }\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi-2\Omega^{2}\underline{\eta}|\mbox{$\nabla\mkern-13.0mu/$ }\phi|^{2}.

Recall that we write the equation D¯K¯𝐷𝐾\underline{D}K in the form:

D¯(K1|u|2)+32Ωtrχ¯(K1|u|2)=¯𝐷𝐾1superscript𝑢232Ωtr¯𝜒𝐾1superscript𝑢2\underline{D}(K-\frac{1}{|u|^{2}})+\frac{3}{2}\Omega\mathrm{tr}\underline{\chi}(K-\frac{1}{|u|^{2}})=\cdots

in order to obtain good enough estimates. However, we have two terms which are still not good enough:

(Ωtrχ¯+2|u|)1|u|2,12Ωtrχ¯μ.Ωtr¯𝜒2𝑢1superscript𝑢212Ωtr¯𝜒𝜇(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|})\frac{1}{|u|^{2}},\ \frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\mu.

Fortunately, these two terms have better bounds after taking derivatives, in view of two improved estimates (4.37), (4.59) for Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} and μ𝜇\mu respectively. So we will firstly estimate the derivatives of K|u|2𝐾superscript𝑢2K-|u|^{-2}, σˇˇ𝜎\check{\sigma}, Ωβ¯+L¯ϕ/ ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi.

Write

D¯(|u|4(|(|u|/ )i(K|u|2)|2+|(|u|/ )iσˇ|2)dμg/)\displaystyle\underline{D}(|u|^{4}(|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2})|^{2}+|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}\check{\sigma}|^{2})\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})
+D(Ω2|u|4|(|u|/ )i(Ωβ¯+L¯ϕ/ ϕ)|2dμg/)\displaystyle+D(\Omega^{-2}|u|^{4}|(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)|^{2}\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}})
=\displaystyle= |u|4+2i/ A(/ B1Bii(Ωβ¯+L¯ϕ/ ϕ)A/ i,B1Bi(K|u|2)\displaystyle|u|^{4+2i}\mbox{$\nabla\mkern-13.0mu/$ }^{A}(\mbox{$\nabla\mkern-13.0mu/$ }^{i}_{B_{1}\cdots B_{i}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)_{A}\mbox{$\nabla\mkern-13.0mu/$ }^{i,B_{1}\cdots B_{i}}(K-|u|^{-2})
+/ B1Bii(ΩβLϕ/ ϕ)A/ i,B1Biσˇ)+|u|4τ4\displaystyle+\mbox{$\nabla\mkern-13.0mu/$ }^{i}_{B_{1}\cdots B_{i}}{}^{*}(\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)_{A}\mbox{$\nabla\mkern-13.0mu/$ }^{i,B_{1}\cdots B_{i}}\check{\sigma})+|u|^{4}\tau_{4}

for 1i41𝑖41\leq i\leq 4, where τ4subscript𝜏4\tau_{4} contains no fifth order derivative of curvature components. By divergence theorem, we have

Ω02(u)δ|u|3(|u|/ )i(K|u|2,σˇ)𝕃u¯2𝕃2(u)2+Ω02(u)Ω1|u|72(|u|/ )i(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]2𝕃2(u¯)2superscriptsubscriptΩ02𝑢𝛿superscriptsubscriptnormsuperscript𝑢3superscript𝑢/ 𝑖𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2¯𝑢superscript𝕃2𝑢2superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscriptΩ1superscript𝑢72superscript𝑢/ 𝑖Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript𝕃2¯𝑢2\displaystyle\Omega_{0}^{2}(u)\delta\||u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{L}^{2}(u)}^{2}+\Omega_{0}^{2}(u)\|\Omega^{-1}|u|^{\frac{7}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{L}^{2}(\underline{u})}^{2}
less-than-or-similar-to\displaystyle\lesssim Ω02(u)δ|u|3(|u|/ )i(K|u|2,σˇ)𝕃u¯2𝕃2(u0)2+Ω02(u)Ω1|u|72(|u|/ )i(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]2𝕃4(0)2superscriptsubscriptΩ02𝑢𝛿superscriptsubscriptnormsuperscript𝑢3superscript𝑢/ 𝑖𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2¯𝑢superscript𝕃2subscript𝑢02superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscriptΩ1superscript𝑢72superscript𝑢/ 𝑖Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript𝕃402\displaystyle\Omega_{0}^{2}(u)\delta\||u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{L}^{2}(u_{0})}^{2}+\Omega_{0}^{2}(u)\|\Omega^{-1}|u|^{\frac{7}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{L}^{4}(0)}^{2}
+0δdu¯u0uduSu¯,uΩ02(u)|u|4|τ4|dμg/\displaystyle+\int_{0}^{\delta}\mathrm{d}\underline{u}^{\prime}\int_{u_{0}}^{u}\mathrm{d}u^{\prime}\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u|^{4}|\tau_{4}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

where

Su¯,uΩ02(u)|u|4|τ4|dμg/Su¯,uΩ02(u)|u|4|τ4,1|dμg/+Su¯,uΩ02(u)|u|4|τ4,2|dμg/\displaystyle\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}|^{4}|\tau_{4}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}|^{4}|\tau_{4,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}+\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}|^{4}|\tau_{4,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}

and the multiplier terms

Su¯,uΩ02(u)|u|4|τ4,1|dμg/Ω02(u)Ω02(u)[|u|6Ωβ¯+L¯ϕ/ ϕ42(|ω|+Ωtrχ,Ωχ^4)+|u|6Ω02|u|1K|u|2,σˇ4Ωβ¯+L¯ϕ/ ϕ4+|u|6Ω02Ωβ¯+Lϕ/ ϕ4η,η¯4K,K|u|2,σˇ4+|u|6Ωβ¯+L¯ϕ/ ϕ4|u|1((|u|/ )(Ωχ^)4Ωχ¯^4+Ωχ^4(|u|/ )(Ωχ¯^)4)+|u|6Ωβ¯+L¯ϕ/ ϕ4|u|1((|u|/ )(Ωtrχ)4Ωtrχ¯4+Ωtrχ4(|u|/ )(Ωtrχ¯)4)+|u|6Ωβ¯+L¯ϕ/ ϕ4(η¯4Ωχ^,Ωtrχ4Ωχ¯^,Ωtrχ¯4)+|u|6Ωβ¯+L¯ϕ/ ϕ4Ωχ¯^4ΩβLϕ/ ϕ4+|u|6Ω02Ωβ¯+L¯ϕ/ ϕ4/ ϕ4|u|1/ ϕ5+|u|6Ωβ¯+L¯ϕ/ ϕ4(Ωχ¯^,Ωtrχ¯4/ ϕ4Lϕ4+Ω02η¯4/ ϕ42)]+|u|6Ω02K|u|2,σˇ4Ωχ¯^,Ωtrχ¯+2|u|14(K|u|2,σˇ4+|u|1η5)+|u|6Ω02K|u|2,σˇ4Ωχ¯^,Ωtrχ¯4η42+|u|6Ω02σˇ4|u|1(|u|/ )L¯ϕ4/ ϕ4+|u|6Ω02K|u|24((|u|2+μ3)(|u|/ )(Ωtrχ¯)3+Ωtrχ¯4(|u|/ )μ3)\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}|^{4}|\tau_{4,1}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&\Omega_{0}^{2}(u)\Omega_{0}^{-2}(u^{\prime})\left[|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}^{2}(|\omega|+\|\Omega\mathrm{tr}\chi,\Omega\widehat{\chi}\|_{4})\right.\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\cdot|u^{\prime}|^{-1}\|K-|u|^{-2},\check{\sigma}\|_{4}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\|\Omega\underline{\beta}+L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\eta,\underline{\eta}\|_{4}\|K,K-|u|^{-2},\check{\sigma}\|_{4}\\ &+|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}|u^{\prime}|^{-1}(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\chi})\|_{4}\|\Omega\widehat{\underline{\chi}}\|_{4}+\|\Omega\widehat{\chi}\|_{4}(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\underline{\chi}})\|_{4})\\ &+|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}|u^{\prime}|^{-1}(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{4}\|\Omega\mathrm{tr}\underline{\chi}\|_{4}+\|\Omega\mathrm{tr}\chi\|_{4}\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{4})\\ &+|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|\underline{\eta}\|_{4}\|\Omega\widehat{\chi},\Omega\mathrm{tr}\chi\|_{4}\|\Omega\widehat{\underline{\chi}},\Omega\mathrm{tr}\underline{\chi}\|_{4}\right)\\ &+|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\Omega\widehat{\underline{\chi}}\|_{4}\|\Omega\beta-L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}|u^{\prime}|^{-1}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{5}\\ &\left.+|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|\Omega\widehat{\underline{\chi}},\Omega\mathrm{tr}\underline{\chi}\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\|L\phi\|_{4}+\Omega_{0}^{2}\|\underline{\eta}\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}^{2}\right)\right]\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\|K-|u|^{-2},\check{\sigma}\|_{4}\|\Omega\widehat{\underline{\chi}},\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1}\|_{4}(\|K-|u|^{-2},\check{\sigma}\|_{4}+|u^{\prime}|^{-1}\|\eta\|_{5})\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\|K-|u|^{-2},\check{\sigma}\|_{4}\|\|\Omega\widehat{\underline{\chi}},\Omega\mathrm{tr}\underline{\chi}\|_{4}\|\eta\|_{4}^{2}\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\|\check{\sigma}\|_{4}|u^{\prime}|^{-1}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\underline{L}\phi\|_{4}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}\|K-|u|^{-2}\|_{4}((|u|^{-2}+\|\mu\|_{3})\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{3}+\|\Omega\mathrm{tr}\underline{\chi}\|_{4}\|(|u|\mbox{$\nabla\mkern-13.0mu/$ })\mu\|_{3})\end{split} (4.83)

and commutation terms

Su¯,uΩ02(u)|u|4|τ4,2|dμg/Ω02(u)Ω02(u)|u|6Ωβ¯+L¯ϕ/ ϕ4((|u|/ )(Ωχ)3Ωβ¯+L¯ϕ/ ϕ3+Ω02(u)|u|K2K|u|2,σˇ4)+|u|6Ω02(u)K|u|2,σˇ4((|u|/ )(Ωχ¯)2K|u|2,σˇ3+|u|K3Ωβ¯+L¯ϕ/ ϕ4).\begin{split}&\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}|^{4}|\tau_{4,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\\ \lesssim&\Omega_{0}^{2}(u)\Omega_{0}^{-2}(u^{\prime})|u^{\prime}|^{6}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\chi)\|_{3}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{3}+\Omega_{0}^{2}(u^{\prime})|u^{\prime}|\|K\|_{2}\|K-|u|^{-2},\check{\sigma}\|_{4}\right)\\ &+|u^{\prime}|^{6}\Omega_{0}^{2}(u^{\prime})\|K-|u|^{-2},\check{\sigma}\|_{4}\left(\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\underline{\chi})\|_{2}\|K-|u|^{-2},\check{\sigma}\|_{3}+|u^{\prime}|\|K\|_{3}\|\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{4}\right).\end{split} (4.84)

(4.83) and (4.84) should be estimated in 𝕃u¯1Lu1superscriptsubscript𝕃¯𝑢1subscriptsuperscript𝐿1𝑢\mathbb{L}_{\underline{u}}^{1}L^{1}_{u}. In the following estimate, Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)4subscriptnormsuperscriptsubscriptΩ01superscript𝑢72Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ4\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{4} should be placed in 𝕃u¯𝕃[u0,u]2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]}. We remark that all terms of the last line of (4.83) contain a factor that is either the derivatives of Ωtrχ¯Ωtr¯𝜒\Omega\mathrm{tr}\underline{\chi} or the derivatives of μ𝜇\mu, which have better estimates. This is because we are considering the derivatives of K|u|2𝐾superscript𝑢2K-|u|^{-2}, σˇˇ𝜎\check{\sigma} and Ωβ¯+L¯ϕ/ ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi but not themselves.

We begin the estimates. The \engordnumber1 line of the right hand side of (4.84) is estimated by,

δ|u|1C34δ332𝒜3C14𝒲12𝒜C1δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢1superscript𝐶34superscript𝛿3superscript3superscript2superscript𝒜3superscript𝐶14superscript𝒲12𝒜less-than-or-similar-tosuperscript𝐶1superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-1}\cdot C^{\frac{3}{4}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-1}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber2 line is estimated by,

δ|u|12C38δ3232𝒜32C38δ3232𝒜32C14δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶38superscript𝛿32superscript32superscript𝒜32less-than-or-similar-tosuperscript𝐶14superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\lesssim C^{-\frac{1}{4}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

To estimate the \engordnumber3 line, we need a refined version of the estimate (4.45) for K𝐾K:

Ω0K𝕃u¯24(u¯)Ω0|u|2+C38δ|u|332𝒜32Ω0|u|2+C32|u|212𝒜12.less-than-or-similar-tosubscriptΩ0subscriptnorm𝐾subscriptsuperscript𝕃2¯𝑢superscript4¯𝑢subscriptΩ0superscript𝑢2superscript𝐶38𝛿superscript𝑢3superscript32superscript𝒜32less-than-or-similar-tosubscriptΩ0superscript𝑢2superscript𝐶32superscript𝑢2superscript12superscript𝒜12\displaystyle\Omega_{0}\|K\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{H}^{4}(\underline{u})}\lesssim\Omega_{0}|u|^{-2}+C^{\frac{3}{8}}\delta|u|^{-3}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\lesssim\Omega_{0}|u|^{-2}+C^{-\frac{3}{2}}|u|^{-2}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}.

Then the \engordnumber3 line is estimated by

less-than-or-similar-to\displaystyle\lesssim Ω0δ|u|12C38δ3232𝒜32C14δ𝒜[Ω0+C3212𝒜12]subscriptΩ0𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶14𝛿𝒜delimited-[]subscriptΩ0superscript𝐶32superscript12superscript𝒜12\displaystyle\Omega_{0}\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot[\Omega_{0}+C^{-\frac{3}{2}}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}]
less-than-or-similar-to\displaystyle\lesssim C18δ3522𝒜52+C34δ332𝒜3C18δ332𝒜3.less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript52superscript2superscript𝒜52superscript𝐶34superscript𝛿3superscript3superscript2superscript𝒜3superscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{\frac{5}{2}}\mathscr{E}^{2}\mathcal{A}^{\frac{5}{2}}+C^{-\frac{3}{4}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber1 term of the \engordnumber4 line is estimated by, placing |u|(|u|/ )(Ωχ^)4superscript𝑢subscriptnormsuperscript𝑢/ Ω^𝜒4|u^{\prime}|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\chi})\|_{4} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢superscriptsubscript𝕃¯𝑢2\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}_{\underline{u}}^{2},

δ|u|12C38δ3232𝒜32C14𝒜C14δ𝒜C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶14𝒜superscript𝐶14𝛿𝒜less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber1 term of the \engordnumber5 line is estimated by, placing |u|(|u|/ )(Ωtrχ)4superscript𝑢subscriptnormsuperscript𝑢/ Ωtr𝜒4|u^{\prime}|\|(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\chi)\|_{4} in 𝕃u¯𝕃[u0,u]superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{\infty}_{[u_{0},u]}

δ|u|12C38δ3232𝒜32C12δ2𝒜21C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶12𝛿superscript2superscript𝒜21less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{2}}\delta\mathscr{F}^{2}\mathscr{E}\mathcal{A}^{2}\cdot 1\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber2 terms of the \engordnumber4 and \engordnumber5 lines are estimated by, placing Ω01|u|32(|u|/ )(Ωχ¯^)5subscriptnormsuperscriptsubscriptΩ01superscriptsuperscript𝑢32superscript𝑢/ Ω^¯𝜒5\|\Omega_{0}^{-1}|u^{\prime}|^{\frac{3}{2}}(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\widehat{\underline{\chi}})\|_{5} in 𝕃u¯𝕃[u0,u]2superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{2}_{[u_{0},u]}, and |u|2(|u|/ )(Ωtrχ¯)5subscriptnormsuperscriptsuperscript𝑢2superscript𝑢/ Ωtr¯𝜒5\||u^{\prime}|^{2}(|u^{\prime}|\mbox{$\nabla\mkern-13.0mu/$ })(\Omega\mathrm{tr}\underline{\chi})\|_{5} in 𝕃u¯𝕃[u0,u]superscriptsubscript𝕃¯𝑢subscriptsuperscript𝕃subscript𝑢0𝑢\mathbb{L}_{\underline{u}}^{\infty}\mathbb{L}^{\infty}_{[u_{0},u]},

δ|u|12C38δ3232𝒜32C14𝒜C14δ𝒲12𝒜C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶14𝒜superscript𝐶14𝛿superscript𝒲12𝒜less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber6 line is estimated by

δ|u|12C38δ3232𝒜32C14δ𝒲12𝒜C14𝒜1C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14𝒜1less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\cdot 1\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber7 line is estimated by,

δ|u|12C38δ3232𝒜32C14δ𝒜C14𝒜C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶14𝛿𝒜superscript𝐶14𝒜less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber8 line is estimated by, placing Ω0|u|32/ ϕ5subscriptΩ0superscript𝑢32subscriptnorm/ italic-ϕ5\Omega_{0}|u|^{\frac{3}{2}}\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{5} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢superscriptsubscript𝕃¯𝑢2\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}_{\underline{u}}^{2},

Ω0δ|u|1C38δ3232𝒜32C14δ𝒜C14δ12𝒜C18δ332𝒜3.less-than-or-similar-toabsentsubscriptΩ0𝛿superscript𝑢1superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscript𝐶14𝛿𝒜superscript𝐶14superscript𝛿12𝒜less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\Omega_{0}\delta|u|^{-1}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber1 term of the \engordnumber9 line is estimated by,

δ|u|12C38δ3232𝒜321C14δ𝒜C14𝒜C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38superscript𝛿32superscript32superscript𝒜321superscript𝐶14𝛿𝒜superscript𝐶14𝒜less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot 1\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\mathscr{F}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber2 term of the \engordnumber9 line is estimated by,

Ω02δ|u|52C38δ3232𝒜32(C14δ𝒲12𝒜)3C1δ332𝒜3.less-than-or-similar-toabsentsuperscriptsubscriptΩ02𝛿superscript𝑢52superscript𝐶38superscript𝛿32superscript32superscript𝒜32superscriptsuperscript𝐶14𝛿superscript𝒲12𝒜3less-than-or-similar-tosuperscript𝐶1superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\Omega_{0}^{2}\delta|u|^{-\frac{5}{2}}\cdot C^{\frac{3}{8}}\delta^{\frac{3}{2}}\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot(C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A})^{3}\lesssim C^{-1}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber10 line is estimated by, placing Ω0|u|1η5subscriptΩ0superscript𝑢1subscriptnorm𝜂5\Omega_{0}|u|^{-1}\|\eta\|_{5} in 𝕃[u0,u]𝕃u¯2subscriptsuperscript𝕃subscript𝑢0𝑢subscriptsuperscript𝕃2¯𝑢\mathbb{L}^{\infty}_{[u_{0},u]}\mathbb{L}^{2}_{\underline{u}},

δ|u|12C38δ32𝒜32C14δ𝒲12𝒜C14δ12𝒜C18δ332𝒜3.less-than-or-similar-toabsent𝛿superscript𝑢12superscript𝐶38𝛿superscript32superscript𝒜32superscript𝐶14𝛿superscript𝒲12𝒜superscript𝐶14superscript𝛿12𝒜less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot C^{\frac{1}{4}}\delta^{\frac{1}{2}}\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber11 line is estimated by

Ω0δ|u|1C38δ32𝒜321(C14δ𝒜)2C18δ332𝒜3.less-than-or-similar-toabsentsubscriptΩ0𝛿superscript𝑢1superscript𝐶38𝛿superscript32superscript𝒜321superscriptsuperscript𝐶14𝛿𝒜2less-than-or-similar-tosuperscript𝐶18superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\Omega_{0}\delta|u|^{-1}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot 1\cdot(C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A})^{2}\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The \engordnumber12 line is estimated by, placing Ω01|u|34L¯ϕ5subscriptnormsuperscriptsubscriptΩ01superscript𝑢34¯𝐿italic-ϕ5\|\Omega_{0}^{-1}|u|^{\frac{3}{4}}\underline{L}\phi\|_{5} in 𝕃u¯𝕃[u0,u]2subscriptsuperscript𝕃¯𝑢subscriptsuperscript𝕃2subscript𝑢0𝑢\mathbb{L}^{\infty}_{\underline{u}}\mathbb{L}^{2}_{[u_{0},u]},

Ω0δ|u|1C38δ32𝒜32C14δ𝒜C14δ𝒜C1δ332𝒜3.less-than-or-similar-toabsentsubscriptΩ0𝛿superscript𝑢1superscript𝐶38𝛿superscript32superscript𝒜32superscript𝐶14𝛿𝒜superscript𝐶14𝛿𝒜less-than-or-similar-tosuperscript𝐶1superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\lesssim\Omega_{0}\delta|u|^{-1}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\cdot C^{\frac{1}{4}}\delta\mathscr{F}\mathscr{E}\mathcal{A}\lesssim C^{-1}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

To estimate the last line, we should use two improved estimates (4.37) and (4.59). From (4.55), we know μ3C2Ω01|u|212𝒜12less-than-or-similar-tosubscriptnorm𝜇3superscript𝐶2superscriptsubscriptΩ01superscript𝑢2superscript12superscript𝒜12\|\mu\|_{3}\lesssim C^{-2}\Omega_{0}^{-1}|u|^{-2}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}, then the last line is then estimated by

δ|u|12C38δ32𝒜32\displaystyle\lesssim\delta|u|^{-\frac{1}{2}}\cdot C^{\frac{3}{8}}\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}\cdot (C212𝒜12C14δ32𝒲12𝒜\displaystyle(C^{-2}\mathscr{F}^{\frac{1}{2}}\mathscr{E}\mathcal{A}^{\frac{1}{2}}\cdot C^{\frac{1}{4}}\delta^{\frac{3}{2}}\mathscr{F}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}
+1C12δ322𝒲12𝒜2)C18δ332𝒜3.\displaystyle+1\cdot C^{\frac{1}{2}}\delta^{\frac{3}{2}}\mathscr{F}^{2}\mathscr{E}\mathscr{W}^{\frac{1}{2}}\mathcal{A}^{2})\lesssim C^{-\frac{1}{8}}\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

The commutation terms (4.84) can be estimated similarly to the estimate of (4.81), using (4.53) for the Gauss curvature K𝐾K. It is easily find that

0δdu¯u0uduSu¯,uΩ02(u)|u|4|τ4,2|dμg/δ332𝒜3.\displaystyle\int_{0}^{\delta}\mathrm{d}\underline{u}^{\prime}\int_{u_{0}}^{u}\mathrm{d}u^{\prime}\int_{S_{\underline{u}^{\prime},u^{\prime}}}\Omega_{0}^{2}(u)|u^{\prime}|^{4}|\tau_{4,2}|\mathrm{d}\mu_{\mbox{$g\mkern-9.0mu/$}}\lesssim\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}.

Therefore, since β¯(u¯=0)=0¯𝛽¯𝑢00\underline{\beta}(\underline{u}=0)=0, for 1i41𝑖41\leq i\leq 4,

Ω02(u)δ|u|3(|u|/ )i(K|u|2,σˇ)𝕃u¯2𝕃2(u)2+Ω02(u)Ω02|u|72(|u|/ )i(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]2𝕃2(u¯)2superscriptsubscriptΩ02𝑢𝛿superscriptsubscriptnormsuperscript𝑢3superscript𝑢/ 𝑖𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2¯𝑢superscript𝕃2𝑢2superscriptsubscriptΩ02𝑢superscriptsubscriptnormsuperscriptsubscriptΩ02superscript𝑢72superscript𝑢/ 𝑖Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript𝕃2¯𝑢2\displaystyle\Omega_{0}^{2}(u)\delta\||u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{L}^{2}(u)}^{2}+\Omega_{0}^{2}(u)\|\Omega_{0}^{-2}|u|^{\frac{7}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{L}^{2}(\underline{u})}^{2}
less-than-or-similar-to\displaystyle\lesssim Ω02(u)δ|u|3(|u|/ )i(K|u|2,σˇ)𝕃u¯2𝕃2(u0)2+δ332𝒜3superscriptsubscriptΩ02𝑢𝛿superscriptsubscriptnormsuperscript𝑢3superscript𝑢/ 𝑖𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2¯𝑢superscript𝕃2subscript𝑢02superscript𝛿3superscript3superscript2superscript𝒜3\displaystyle\Omega_{0}^{2}(u)\delta\||u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{L}^{2}(u_{0})}^{2}+\delta^{3}\mathscr{F}^{3}\mathscr{E}^{2}\mathcal{A}^{3}

which means

Ω0|u|3(|u|/ )i(K|u|2,σˇ)𝕃u¯2𝕃2(u)+Ω0δ12Ω01|u|72(|u|/ )i(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]2𝕃2(u¯)δ32𝒜32.less-than-or-similar-tosubscriptΩ0subscriptdelimited-∥∥superscript𝑢3superscript𝑢/ 𝑖𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2¯𝑢superscript𝕃2𝑢subscriptΩ0superscript𝛿12subscriptdelimited-∥∥superscriptsubscriptΩ01superscript𝑢72superscript𝑢/ 𝑖Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript𝕃2¯𝑢𝛿superscript32superscript𝒜32\begin{split}&\Omega_{0}\||u|^{3}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{L}^{2}(u)}\\ &+\Omega_{0}\delta^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(|u|\mbox{$\nabla\mkern-13.0mu/$ })^{i}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{L}^{2}(\underline{u})}\lesssim\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}.\end{split} (4.85)

It remains to estimate the zeroth order bound of K𝐾K and Ωβ¯+L¯ϕ/ ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi, that is, to show

Ω0|u|3(K|u|2,σˇ)𝕃u¯2𝕃2(u)+Ω0δ12Ω01|u|72(Ωβ¯+L¯ϕ/ ϕ)𝕃[u0,u]2𝕃2(u¯)δ32𝒜32.less-than-or-similar-tosubscriptΩ0subscriptnormsuperscript𝑢3𝐾superscript𝑢2ˇ𝜎subscriptsuperscript𝕃2¯𝑢superscript𝕃2𝑢subscriptΩ0superscript𝛿12subscriptnormsuperscriptsubscriptΩ01superscript𝑢72Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕsubscriptsuperscript𝕃2subscript𝑢0𝑢superscript𝕃2¯𝑢𝛿superscript32superscript𝒜32\displaystyle\Omega_{0}\||u|^{3}(K-|u|^{-2},\check{\sigma})\|_{\mathbb{L}^{2}_{\underline{u}}\mathbb{L}^{2}(u)}+\Omega_{0}\delta^{-\frac{1}{2}}\|\Omega_{0}^{-1}|u|^{\frac{7}{2}}(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi)\|_{\mathbb{L}^{2}_{[u_{0},u]}\mathbb{L}^{2}(\underline{u})}\lesssim\delta\mathscr{F}^{\frac{3}{2}}\mathscr{E}\mathcal{A}^{\frac{3}{2}}.

For K𝐾K, this is followed from (4.52) and Ω01subscriptΩ01\Omega_{0}\leq 1. For σˇˇ𝜎\check{\sigma} and Ωβ¯+L¯ϕ/ ϕΩ¯𝛽¯𝐿italic-ϕ/ italic-ϕ\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi, this is followed from directly integrating the Bianchi equation for D¯σˇ¯𝐷ˇ𝜎\underline{D}\check{\sigma} and D(Ωβ¯+L¯ϕ/ ϕ)𝐷Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕD(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi). The reason for why this works is simple. Remember that the zeroth order quantity cannot be estimated because we have two bad terms |u|2(Ωtrχ¯+2|u|1)superscript𝑢2Ωtr¯𝜒2superscript𝑢1|u|^{-2}(\Omega\mathrm{tr}\underline{\chi}+2|u|^{-1}) and 12Ωtrχ¯μ12Ωtr¯𝜒𝜇\frac{1}{2}\Omega\mathrm{tr}\underline{\chi}\mu whose estimates are good enough only after we take derivatives on them. However, both of these two terms come from the equation for D¯(K|u|2)¯𝐷𝐾superscript𝑢2\underline{D}(K-|u|^{-2}). This means when we directly integrate the equations for D¯σˇ¯𝐷ˇ𝜎\underline{D}\check{\sigma} and D(Ωβ¯+L¯ϕ/ ϕ)𝐷Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕD(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi), we will not encounter these two terms. The price is that we will lose derivative because we will encounter curl/ (Ωβ¯+L¯ϕ/ ϕ)curl/ Ω¯𝛽¯𝐿italic-ϕ/ italic-ϕ\mbox{$\mathrm{curl}\mkern-13.0mu/$ }(\Omega\underline{\beta}+\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi) and / K,/ σˇ/ 𝐾/ ˇ𝜎\mbox{$\nabla\mkern-13.0mu/$ }K,\mbox{$\nabla\mkern-13.0mu/$ }\check{\sigma}. But this still works because we are only dealing with the zeroth order.

Finally, we have completed the proof of Proposition 4.5.

 

Combining all the propositions proved above then concludes the proof of Theorem 3.1.

5. Formation of trapped surfaces

In this section, we prove a formation of trapped surfaces theorem. The theorem we will prove is the following:

Theorem 5.1 (Formation of trapped surfaces).

There exists a universal constant C1C0subscript𝐶1subscript𝐶0C_{1}\geq C_{0} such that the following statement is true. Consider the same characteristic initial value problem as in Theorem 3.1 and let \mathscr{F}, 𝒜𝒜\mathcal{A}, \mathscr{E} be defined as in Theorem 3.1 with three numbers δ𝛿\delta, u0<u1<0subscript𝑢0subscript𝑢10u_{0}<u_{1}<0. Suppose that the initial data given on C¯0subscript¯𝐶0\underline{C}_{0} is spherically symmetric with Ω(u0)1Ωsubscript𝑢01\Omega(u_{0})\leq 1, and

𝒜a𝒜𝑎\displaystyle\mathcal{A}\leq a (5.1)

for some constant a1𝑎1a\geq 1. Suppose also that CC1𝐶subscript𝐶1C\geq C_{1} and

Ω02(u0)aC2Ω04(u0)2.superscriptsubscriptΩ02subscript𝑢0𝑎superscript𝐶2superscriptsubscriptΩ04subscript𝑢0superscript2\displaystyle\Omega_{0}^{2}(u_{0})\mathscr{F}a\geq C^{-2}\Omega_{0}^{4}(u_{0})\mathscr{E}^{2}. (5.2)

Then the smooth solution of the Einstein-scalar field equations exists for 0u¯δ0¯𝑢𝛿0\leq\underline{u}\leq\delta, u0uu1subscript𝑢0𝑢subscript𝑢1u_{0}\leq u\leq u_{1}, where u1subscript𝑢1u_{1} is defined by

Ω02(u1)|u1|=C2Ω02(u0)δa.subscriptsuperscriptΩ20subscript𝑢1subscript𝑢1superscript𝐶2superscriptsubscriptΩ02subscript𝑢0𝛿𝑎\Omega^{2}_{0}(u_{1})|u_{1}|=C^{2}\Omega_{0}^{2}(u_{0})\delta\mathscr{F}a. (5.3)

If in addition

infϑS20δ|u0|2(|Ωχ^|2+2|Lϕ|2)(u¯,u0,ϑ)du¯17C2Ω02(u0)δa,subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0𝛿superscriptsubscript𝑢02superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢0italic-ϑdifferential-dsuperscript¯𝑢17superscript𝐶2superscriptsubscriptΩ02subscript𝑢0𝛿𝑎\inf_{\vartheta\in S^{2}}\int_{0}^{\delta}|u_{0}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u}^{\prime},u_{0},\vartheta)\mathrm{d}\underline{u}^{\prime}\geq 17C^{2}\Omega_{0}^{2}(u_{0})\delta\mathscr{F}a, (5.4)

together with

Ω02(u0)a16C2(u0u1Ω02h|ψ||u|du)2,superscriptsubscriptΩ02subscript𝑢0𝑎16superscript𝐶2superscriptsuperscriptsubscriptsubscript𝑢0subscript𝑢1superscriptsubscriptΩ02𝜓superscript𝑢differential-dsuperscript𝑢2\displaystyle\Omega_{0}^{2}(u_{0})\mathscr{F}a\geq 16C^{-2}\left(\int_{u_{0}}^{u_{1}}\frac{\Omega_{0}^{2}h|\psi|}{|u^{\prime}|}\mathrm{d}u^{\prime}\right)^{2}, (5.5)

hold, then the sphere Sδ,u1subscript𝑆𝛿subscript𝑢1S_{\delta,u_{1}} is a closed trapped surface.

Proof.

First of all, we shall prove the solution exists in the region 0u¯δ0¯𝑢𝛿0\leq\underline{u}\leq\delta, u0uu1subscript𝑢0𝑢subscript𝑢1u_{0}\leq u\leq u_{1} by applying Theorem 3.1. We should verify δ𝛿\delta and u1subscript𝑢1u_{1} defined by (5.3) satisfies the smallness and auxiliary conditions (3.4) and (3.5). By (5.1), (5.2) and (5.3), we have

max{Ω02(u0)δ|u1|12,C2δ|u1|1𝒜}C2δ|u1|1aΩ02(u1)Ω02(u0)𝒲1,superscriptsubscriptΩ02subscript𝑢0𝛿superscriptsubscript𝑢11superscript2superscript𝐶2𝛿superscriptsubscript𝑢11𝒜superscript𝐶2𝛿superscriptsubscript𝑢11𝑎superscriptsubscriptΩ02subscript𝑢1superscriptsubscriptΩ02subscript𝑢0superscript𝒲1\displaystyle\max\left\{\Omega_{0}^{2}(u_{0})\delta|u_{1}|^{-1}\mathscr{E}^{2},C^{2}\delta|u_{1}|^{-1}\mathscr{F}\mathcal{A}\right\}\leq C^{2}\delta|u_{1}|^{-1}\mathscr{F}a\leq\frac{\Omega_{0}^{2}(u_{1})}{\Omega_{0}^{2}(u_{0})}\leq\mathscr{W}^{-1},

where we use the fact that x|logx|1𝑥𝑥1x|\log x|\leq 1 for x(0,1]𝑥01x\in(0,1]. This verifies both the smallness conditions and the auxiliary condition, if CC1C0𝐶subscript𝐶1subscript𝐶0C\geq C_{1}\geq C_{0}. As a consequence, the estimates stated in Theorem 3.1 also hold.

The next step is to improve the estimates for η𝜂\eta and / ϕ/ italic-ϕ\mbox{$\nabla\mkern-13.0mu/$ }\phi for up to the third order derivatives. We will prove

Proposition 5.1.

Suppose that the assumptions and conclusions in Theorem 3.1 hold, with CC1C0𝐶subscript𝐶1subscript𝐶0C\geq C_{1}\geq C_{0}. Then if C1subscript𝐶1C_{1} is sufficiently large,

/ ϕ3(u¯,u),η3(u¯,u),𝒲12η¯3(u¯,u)δ|u|2𝒜.less-than-or-similar-tosubscriptnorm/ italic-ϕsuperscript3¯𝑢𝑢subscriptnorm𝜂superscript3¯𝑢𝑢superscript𝒲12subscriptnorm¯𝜂superscript3¯𝑢𝑢𝛿superscript𝑢2𝒜\displaystyle\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{3}(\underline{u},u)},\|\eta\|_{\mathbb{H}^{3}(\underline{u},u)},\mathscr{W}^{-\frac{1}{2}}\|\underline{\eta}\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}.
Proof.

From the equation (4.17), we have

/ ϕ3(u¯,u)δ|u|1Lϕ𝕃u¯4(u)δ|u|2𝒜.less-than-or-similar-tosubscriptnorm/ italic-ϕsuperscript3¯𝑢𝑢𝛿superscript𝑢1subscriptnorm𝐿italic-ϕsubscriptsuperscript𝕃¯𝑢superscript4𝑢less-than-or-similar-to𝛿superscript𝑢2𝒜\displaystyle\|\mbox{$\nabla\mkern-13.0mu/$ }\phi\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim\delta|u|^{-1}\|L\phi\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}. (5.6)

We plug two Codazzi equations (4.65) and (4.78) in two transport equations (4.27) and (4.29). We have two equations

Dη=Ωtrχη¯2Lϕ/ ϕ+div/ (Ωχ^)12/ (Ωtrχ).𝐷𝜂Ωtr𝜒¯𝜂2𝐿italic-ϕ/ italic-ϕdiv/ Ω^𝜒12/ Ωtr𝜒\displaystyle D\eta=\Omega\mathrm{tr}\chi\underline{\eta}-2L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\chi})-\frac{1}{2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi). (5.7)
D¯η¯=Ωtrχ¯η2L¯ϕ/ ϕ+div/ (Ωχ¯^)12/ (Ωtrχ¯).¯𝐷¯𝜂Ωtr¯𝜒𝜂2¯𝐿italic-ϕ/ italic-ϕdiv/ Ω^¯𝜒12/ Ωtr¯𝜒\displaystyle\underline{D}\underline{\eta}=\Omega\mathrm{tr}\underline{\chi}\eta-2\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi+\mbox{$\mathrm{div}\mkern-13.0mu/$ }(\Omega\widehat{\underline{\chi}})-\frac{1}{2}\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}). (5.8)

We assume

η¯3(u¯,u)C14δ|u|2𝒲12𝒜.less-than-or-similar-tosubscriptnorm¯𝜂superscript3¯𝑢𝑢superscript𝐶14𝛿superscript𝑢2superscript𝒲12𝒜\displaystyle\|\underline{\eta}\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (5.9)

Then from equation (5.7), we have

η3(u¯,u)δΩtrχη¯,2Lϕ/ ϕ,/ (Ωtrχ)𝕃u¯3(u)+δ|u|1Ωχ^𝕃u¯4(u)δ|u|1𝒜C14δ|u|2𝒲12𝒜+δ|u|2𝒜δ|u|2𝒜.\begin{split}\|\eta\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim&\delta\|\Omega\mathrm{tr}\chi\underline{\eta},2L\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi,\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\chi)\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{3}(u)}+\delta|u|^{-1}\|\Omega\widehat{\chi}\|_{\mathbb{L}^{\infty}_{\underline{u}}\mathbb{H}^{4}(u)}\\ \lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-2}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}+\delta\cdot|u|^{-2}\mathscr{F}\mathcal{A}\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}.\end{split} (5.10)

Then from the equation (5.8) and the above estimate (5.10), we have

|u|η¯3(u¯,u)|u|2(Ωtrχ¯η,L¯ϕ/ ϕ,/ (Ωtrχ¯))𝕃[u0,u]13(u¯)+|u|Ωχ¯^𝕃[u0,u]14(u¯)δ|u|1𝒲12𝒜.less-than-or-similar-tosubscriptdelimited-∥∥𝑢¯𝜂superscript3¯𝑢𝑢subscriptdelimited-∥∥superscript𝑢2Ωtr¯𝜒𝜂¯𝐿italic-ϕ/ italic-ϕ/ Ωtr¯𝜒subscriptsuperscript𝕃1subscript𝑢0𝑢superscript3¯𝑢subscriptdelimited-∥∥𝑢Ω^¯𝜒subscriptsuperscript𝕃1subscript𝑢0𝑢superscript4¯𝑢less-than-or-similar-to𝛿superscript𝑢1superscript𝒲12𝒜\begin{split}\||u|\underline{\eta}\|_{\mathbb{H}^{3}(\underline{u},u)}\lesssim&\||u|^{2}(\Omega\mathrm{tr}\underline{\chi}\eta,\underline{L}\phi\mbox{$\nabla\mkern-13.0mu/$ }\phi,\mbox{$\nabla\mkern-13.0mu/$ }(\Omega\mathrm{tr}\underline{\chi}))\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{3}(\underline{u})}+\||u|\Omega\widehat{\underline{\chi}}\|_{\mathbb{L}^{1}_{[u_{0},u]}\mathbb{H}^{4}(\underline{u})}\\ \lesssim&\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{split} (5.11)

Then this improves (5.9) when C1subscript𝐶1C_{1} is sufficiently large, which implies that the estimates (5.10) and (5.11) hold without assuming (5.9). We have completed the improved estimates.

 

Then we begin to prove that at Sδ,u1subscript𝑆𝛿subscript𝑢1S_{\delta,u_{1}}, trχ,Ωtrχ¯<0trsuperscript𝜒Ωtr¯𝜒0\mathrm{tr}\chi^{\prime},\Omega\mathrm{tr}\underline{\chi}<0. That Ωtrχ¯<0Ωtr¯𝜒0\Omega\mathrm{tr}\underline{\chi}<0 is followed easily by, for any u¯[0,δ],u[u0,u1]formulae-sequence¯𝑢0𝛿𝑢subscript𝑢0subscript𝑢1\underline{u}\in[0,\delta],u\in[u_{0},u_{1}], ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2},

|Ωtrχ¯(u¯,u,ϑ)+2|u||cδ|u|2𝒜cC1|u|1|u|Ωtr¯𝜒¯𝑢𝑢italic-ϑ2𝑢𝑐𝛿superscript𝑢2𝒜𝑐superscript𝐶1𝑢1𝑢\displaystyle\left|\Omega\mathrm{tr}\underline{\chi}(\underline{u},u,\vartheta)+\frac{2}{|u|}\right|\leq c\delta|u|^{-2}\mathscr{F}\mathcal{A}\leq\frac{cC^{-1}}{|u|}\leq\frac{1}{|u|}

if C1subscript𝐶1C_{1} is sufficiently large.

In the rest of the proof, the estimates are derived for any particular ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2} and we only need to prove trχ(δ,u1,ϑ)<0trsuperscript𝜒𝛿subscript𝑢1italic-ϑ0\mathrm{tr}\chi^{\prime}(\delta,u_{1},\vartheta)<0. We integrate the equation (4.25) along the u𝑢u curve at any u[u0,u1]𝑢subscript𝑢0subscript𝑢1u\in[u_{0},u_{1}], we have

|trχ2h|u||δsupu¯|Ωtrχ|supu¯|trχ|+Ω02(u)|u|20δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯.less-than-or-similar-totrsuperscript𝜒2𝑢𝛿subscriptsupremum¯𝑢Ωtr𝜒subscriptsupremum¯𝑢trsuperscript𝜒superscriptsubscriptΩ02𝑢superscript𝑢2superscriptsubscript0𝛿superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle\left|\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|}\right|\lesssim\delta\sup_{\underline{u}}|\Omega\mathrm{tr}\chi|\sup_{\underline{u}}|\mathrm{tr}\chi^{\prime}|+\Omega_{0}^{-2}(u)|u|^{-2}\int_{0}^{\delta}|u|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}. (5.12)

Therefore, we have

|trχ|less-than-or-similar-totrsuperscript𝜒absent\displaystyle|\mathrm{tr}\chi^{\prime}|\lesssim 1|u|+δ|u|1𝒜supu¯|trχ|+Ω02(u)|u|20δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯.1𝑢𝛿superscript𝑢1𝒜subscriptsupremum¯𝑢trsuperscript𝜒superscriptsubscriptΩ02𝑢superscript𝑢2superscriptsubscript0𝛿superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle\frac{1}{|u|}+\delta|u|^{-1}\mathscr{F}\mathcal{A}\sup_{\underline{u}}|\mathrm{tr}\chi^{\prime}|+\Omega_{0}^{-2}(u)|u|^{-2}\int_{0}^{\delta}|u|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}.

If C1subscript𝐶1C_{1} is sufficiently large such that the second term on the right hand side can be absorbed by the left hand side, we have

|trχ|1|u|+Ω02(u)|u|20δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯.less-than-or-similar-totrsuperscript𝜒1𝑢superscriptsubscriptΩ02𝑢superscript𝑢2superscriptsubscript0𝛿superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle|\mathrm{tr}\chi^{\prime}|\lesssim\frac{1}{|u|}+\Omega_{0}^{-2}(u)|u|^{-2}\int_{0}^{\delta}|u|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}. (5.13)

Substitute this back to (5.12) we have

|trχ2h|u||δ|u|2𝒜+Ω02(u)|u|20δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯.less-than-or-similar-totrsuperscript𝜒2𝑢𝛿superscript𝑢2𝒜superscriptsubscriptΩ02𝑢superscript𝑢2superscriptsubscript0𝛿superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle\left|\mathrm{tr}\chi^{\prime}-\frac{2h}{|u|}\right|\lesssim\delta|u|^{-2}\mathscr{F}\mathcal{A}+\Omega_{0}^{-2}(u)|u|^{-2}\int_{0}^{\delta}|u|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}. (5.14)

Now we turn to the equations (4.23) and (4.14), and compute

u(|u|2|Ωχ^|2+2|u|2|Lϕ|2)=2|u|2Ω2(/ ^η+η^η+/ ϕ^/ ϕ)/ b(Ωχ^),Ωχ^+4|u|(Ω2Δ/ ϕ+2Ω2η,/ ϕ/ bLϕ)(|u|Lϕ)(Ωtrχ¯+2|u|)(|u|2|Ωχ^|2+2|u|2|Lϕ|2)Ωtrχ|u|2Ωχ¯^,Ωχ^2Ωtrχ|u|L¯ϕ|u|Lϕ,𝑢superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscript𝐿italic-ϕ22superscript𝑢2superscriptΩ2/ ^tensor-product𝜂𝜂^tensor-product𝜂/ italic-ϕ^tensor-product/ italic-ϕsubscript/ 𝑏Ω^𝜒Ω^𝜒4𝑢superscriptΩ2Δ/ italic-ϕ2superscriptΩ2𝜂/ italic-ϕsubscript/ 𝑏𝐿italic-ϕ𝑢𝐿italic-ϕΩtr¯𝜒2𝑢superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscript𝐿italic-ϕ2Ωtr𝜒superscript𝑢2Ω^¯𝜒Ω^𝜒2Ωtr𝜒𝑢¯𝐿italic-ϕ𝑢𝐿italic-ϕ\begin{split}&\frac{\partial}{\partial u}(|u|^{2}|\Omega\widehat{\chi}|^{2}+2|u|^{2}|L\phi|^{2})=2|u|^{2}\langle\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\eta+\eta\widehat{\otimes}\eta+\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi)-\mbox{$\nabla\mkern-13.0mu/$ }_{b}(\Omega\widehat{\chi}),\Omega\widehat{\chi}\rangle\\ &+4|u|(\Omega^{2}\mbox{$\Delta\mkern-13.0mu/$ }\phi+2\Omega^{2}\langle\eta,\mbox{$\nabla\mkern-13.0mu/$ }\phi\rangle-\mbox{$\nabla\mkern-13.0mu/$ }_{b}L\phi)\cdot(|u|L\phi)-\left(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\right)(|u|^{2}|\Omega\widehat{\chi}|^{2}+2|u|^{2}|L\phi|^{2})\\ &-\Omega\mathrm{tr}\chi|u|^{2}\langle\Omega\widehat{\underline{\chi}},\Omega\widehat{\chi}\rangle-2\Omega\mathrm{tr}\chi|u|\underline{L}\phi\cdot|u|L\phi,\end{split} (5.15)

where b𝑏b is a vector field satisfying Db=4Ω2ζ𝐷𝑏4superscriptΩ2superscript𝜁Db=-4\Omega^{2}\zeta^{\sharp}, which implies, together with (5.10), (5.11), and b(u¯=0)=0𝑏¯𝑢00b(\underline{u}=0)=0,

|b|Ω02δ2|u|2𝒲12𝒜.less-than-or-similar-to𝑏superscriptsubscriptΩ02superscript𝛿2superscript𝑢2superscript𝒲12𝒜\displaystyle|b|\lesssim\Omega_{0}^{2}\delta^{2}|u|^{-2}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}. (5.16)

Using the (5.10), (5.11), (5.16), and the Sobolev inequalities, the first two lines of the right hand side can be bounded in Lsuperscript𝐿L^{\infty} with improved estimates. We then integrate (5.15) along the u¯¯𝑢\underline{u} curve (but not the integral curve of D¯¯𝐷\underline{D}), and the right hand side should be estimated in u0u||du=u0u||(u¯,u,ϑ)du\int_{u_{0}}^{u}|\cdot|\mathrm{d}u^{\prime}=\int_{u_{0}}^{u}|\cdot|(\underline{u},u^{\prime},\vartheta)\mathrm{d}u^{\prime}. The first two terms of the right hand side are estimated by

u0uΩ2(u)δ|u|2𝒜(|u||Ωχ^|+|u||Lϕ|)du.less-than-or-similar-toabsentsuperscriptsubscriptsubscript𝑢0𝑢superscriptΩ2superscript𝑢𝛿superscriptsuperscript𝑢2𝒜𝑢Ω^𝜒𝑢𝐿italic-ϕdifferential-dsuperscript𝑢\displaystyle\lesssim\int_{u_{0}}^{u}\Omega^{2}(u^{\prime})\cdot\delta|u^{\prime}|^{-2}\mathscr{F}\mathcal{A}\cdot(|u||\Omega\widehat{\chi}|+|u||L\phi|)\mathrm{d}u^{\prime}.

The \engordnumber3 term of the right hand side is estimated by

δ𝒜u0u|u|2(|u|2|Ωχ^|2+2|u|2|Lϕ|2)du.less-than-or-similar-toabsent𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2superscriptsuperscript𝑢2superscriptΩ^𝜒22superscriptsuperscript𝑢2superscript𝐿italic-ϕ2differential-dsuperscript𝑢\displaystyle\lesssim\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}(|u^{\prime}|^{2}|\Omega\widehat{\chi}|^{2}+2|u^{\prime}|^{2}|L\phi|^{2})\mathrm{d}u^{\prime}.

The \engordnumber1 term of the \engordnumber3 line is estimated by, using (5.13) to estimate ΩtrχΩtr𝜒\Omega\mathrm{tr}\chi,

less-than-or-similar-to\displaystyle\lesssim δ𝒜u0u|u|1(|u||Ωχ^|)(Ω02(u)|u|+|u|20δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯)du𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1superscript𝑢Ω^𝜒superscriptsubscriptΩ02superscript𝑢superscript𝑢superscriptsuperscript𝑢2superscriptsubscript0𝛿superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢differential-dsuperscript𝑢\displaystyle\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-1}(|u^{\prime}||\Omega\widehat{\chi}|)\left(\frac{\Omega_{0}^{2}(u^{\prime})}{|u^{\prime}|}+|u^{\prime}|^{-2}\int_{0}^{\delta}|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}\right)\mathrm{d}u^{\prime}
less-than-or-similar-to\displaystyle\lesssim δ𝒜Ω02(u0)u0u|u|2|u||Ωχ^|du+δ|u|22𝒜2supu0uu0δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯𝛿𝒜superscriptsubscriptΩ02subscript𝑢0superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2superscript𝑢Ω^𝜒differential-dsuperscript𝑢𝛿superscript𝑢2superscript2superscript𝒜2subscriptsupremumsubscript𝑢0superscript𝑢𝑢superscriptsubscript0𝛿superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle\delta\mathscr{F}\mathcal{A}\Omega_{0}^{2}(u_{0})\int_{u_{0}}^{u}|u^{\prime}|^{-2}\cdot|u^{\prime}||\Omega\widehat{\chi}|\mathrm{d}u^{\prime}+\delta|u|^{-2}\mathscr{F}^{2}\mathcal{A}^{2}\sup_{u_{0}\leq u^{\prime}\leq u}\int_{0}^{\delta}|u^{\prime}|^{2}\left(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2}\right)\mathrm{d}\underline{u}

The last term should be estimated more carefully. We estimate, using (5.13) and (5.14),

u0u|Ωtrχ|u|L¯ϕ|u|Lϕ2Ω2hψ|u||u|Lϕ|dusuperscriptsubscriptsubscript𝑢0𝑢Ωtr𝜒superscript𝑢¯𝐿italic-ϕsuperscript𝑢𝐿italic-ϕ2superscriptΩ2𝜓superscript𝑢superscript𝑢𝐿italic-ϕdifferential-dsuperscript𝑢\displaystyle\int_{u_{0}}^{u}\left|\Omega\mathrm{tr}\chi|u^{\prime}|\underline{L}\phi\cdot|u^{\prime}|L\phi-\frac{2\Omega^{2}h\psi}{|u^{\prime}|}|u^{\prime}|L\phi\right|\mathrm{d}u^{\prime}
less-than-or-similar-to\displaystyle\lesssim u0u|Ωtrχ2Ω2h|u||ψ|u||Lϕ|du+u0u|Ωtrχ|||u|L¯ϕψuLϕ|dusuperscriptsubscriptsubscript𝑢0𝑢Ωtr𝜒2superscriptΩ2superscript𝑢𝜓superscript𝑢𝐿italic-ϕdsuperscript𝑢superscriptsubscriptsubscript𝑢0𝑢Ωtr𝜒superscript𝑢¯𝐿italic-ϕ𝜓normsuperscript𝑢𝐿italic-ϕdifferential-dsuperscript𝑢\displaystyle\int_{u_{0}}^{u}\left|\Omega\mathrm{tr}\chi-\frac{2\Omega^{2}h}{|u^{\prime}|}\right|\cdot\psi|u^{\prime}||L\phi|\mathrm{d}u^{\prime}+\int_{u_{0}}^{u}|\Omega\mathrm{tr}\chi|||u^{\prime}|\underline{L}\phi-\psi||u^{\prime}||L\phi|\mathrm{d}u^{\prime}
less-than-or-similar-to\displaystyle\lesssim Ω02δ𝒜u0u|u|2|ψ||u||Lϕ|du+𝒜u0u|u|2|ψ|dusupu0uu0δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯superscriptsubscriptΩ02𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2𝜓superscript𝑢𝐿italic-ϕdifferential-dsuperscript𝑢𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2𝜓differential-dsuperscript𝑢subscriptsupremumsubscript𝑢0superscript𝑢𝑢superscriptsubscript0𝛿superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle\Omega_{0}^{2}\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}|\psi|\cdot|u^{\prime}||L\phi|\mathrm{d}u^{\prime}+\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}|\psi|\mathrm{d}u^{\prime}\sup_{u_{0}\leq u^{\prime}\leq u}\int_{0}^{\delta}|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}
+Ω02u0u|u|1||u|L¯ϕψ||u||Lϕ|dusuperscriptsubscriptΩ02superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1superscript𝑢¯𝐿italic-ϕ𝜓superscript𝑢𝐿italic-ϕdifferential-dsuperscript𝑢\displaystyle+\Omega_{0}^{2}\int_{u_{0}}^{u}|u^{\prime}|^{-1}||u^{\prime}|\underline{L}\phi-\psi|\cdot|u^{\prime}||L\phi|\mathrm{d}u^{\prime}
+𝒜u0u|u|2||u|L¯ϕψ|dusupu0uu0δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2superscriptsuperscript𝑢¯𝐿italic-ϕ𝜓dsuperscript𝑢subscriptsupremumsubscript𝑢0superscript𝑢𝑢superscriptsubscript0𝛿superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle+\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}||u^{\prime}|\underline{L}\phi-\psi|\mathrm{d}u^{\prime}\sup_{u_{0}\leq u^{\prime}\leq u}\int_{0}^{\delta}|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}
less-than-or-similar-to\displaystyle\lesssim Ω02δ𝒜u0u|u|2|ψ||u||Lϕ|du+|u|1𝒲12𝒜supu0uu0δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯superscriptsubscriptΩ02𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2𝜓superscript𝑢𝐿italic-ϕdifferential-dsuperscript𝑢superscript𝑢1superscript𝒲12𝒜subscriptsupremumsubscript𝑢0superscript𝑢𝑢superscriptsubscript0𝛿superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle\Omega_{0}^{2}\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}|\psi|\cdot|u^{\prime}||L\phi|\mathrm{d}u^{\prime}+|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\sup_{u_{0}\leq u^{\prime}\leq u}\int_{0}^{\delta}|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}
+Ω02u0u|u|1||u|L¯ϕψ||u||Lϕ|du+δ|u|22𝒜2supu0uu0δ|u|2(|Ωχ^|2+2|Lϕ|2)du¯.superscriptsubscriptΩ02superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1superscript𝑢¯𝐿italic-ϕ𝜓superscript𝑢superscript𝐿italic-ϕdsuperscript𝑢𝛿𝑢2superscript2superscript𝒜2subscriptsupremumsubscript𝑢0superscript𝑢𝑢superscriptsubscript0𝛿superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2differential-d¯𝑢\displaystyle+\Omega_{0}^{2}\int_{u_{0}}^{u}|u^{\prime}|^{-1}||u^{\prime}|\underline{L}\phi-\psi|\cdot|u^{\prime}||L\phi|\mathrm{d}u^{\prime}+\delta|u|^{-2}\mathscr{F}^{2}\mathcal{A}^{2}\sup_{u_{0}\leq u^{\prime}\leq u}\int_{0}^{\delta}|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})\mathrm{d}\underline{u}.

Integrating (5.15), using the above all estimates, for some universal c𝑐c,

±{[|u|2|Ωχ^|2+2|u|2|Lϕ|2](u¯,u,ϑ)[|u|2|Ωχ^|2+2|u|2|Lϕ|2](u¯,u0,ϑ)}plus-or-minusdelimited-[]superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscript𝐿italic-ϕ2¯𝑢𝑢italic-ϑdelimited-[]superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscript𝐿italic-ϕ2¯𝑢subscript𝑢0italic-ϑ\displaystyle\pm\left\{[|u|^{2}|\Omega\widehat{\chi}|^{2}+2|u|^{2}|L\phi|^{2}](\underline{u},u,\vartheta)-[|u|^{2}|\Omega\widehat{\chi}|^{2}+2|u|^{2}|L\phi|^{2}](\underline{u},u_{0},\vartheta)\right\}
\displaystyle\leq c|u|1𝒜𝒲12supu0uu0δ[|u|2(|Ωχ^|2+2|Lϕ|2)](u¯,u,ϑ)du¯𝑐superscript𝑢1𝒜superscript𝒲12subscriptsupremumsubscript𝑢0superscript𝑢𝑢superscriptsubscript0𝛿delimited-[]superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢𝑢italic-ϑdifferential-dsuperscript¯𝑢\displaystyle c|u|^{-1}\mathscr{F}\mathcal{A}\mathscr{W}^{\frac{1}{2}}\sup_{u_{0}\leq u^{\prime}\leq u}\int_{0}^{\delta}[|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})](\underline{u}^{\prime},u,\vartheta)\mathrm{d}\underline{u}^{\prime}
+cδ𝒜u0u|u|2(|u|2|Ωχ^|2+2|u|2|Lϕ|2)(u¯,u,ϑ)du+u0u4Ω02h|ψ||u||u||Lϕ|(u¯,u,ϑ)du𝑐𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢2superscriptsuperscript𝑢2superscriptΩ^𝜒22superscriptsuperscript𝑢2superscript𝐿italic-ϕ2¯𝑢superscript𝑢italic-ϑdifferential-dsuperscript𝑢superscriptsubscriptsubscript𝑢0𝑢4superscriptsubscriptΩ02𝜓superscript𝑢superscript𝑢𝐿italic-ϕ¯𝑢superscript𝑢italic-ϑdifferential-dsuperscript𝑢\displaystyle+c\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}(|u^{\prime}|^{2}|\Omega\widehat{\chi}|^{2}+2|u^{\prime}|^{2}|L\phi|^{2})(\underline{u},u^{\prime},\vartheta)\mathrm{d}u^{\prime}+\int_{u_{0}}^{u}\frac{4\Omega_{0}^{2}h|\psi|}{|u^{\prime}|}\cdot|u^{\prime}||L\phi|(\underline{u},u^{\prime},\vartheta)\mathrm{d}u^{\prime}
+cΩ02(u0)δ𝒜u0u|u|2(1+|ψ|)(|u||Lϕ|+|u||Ωχ^|)(u¯,u,ϑ)du𝑐superscriptsubscriptΩ02subscript𝑢0𝛿𝒜superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢21𝜓superscript𝑢𝐿italic-ϕsuperscript𝑢Ω^𝜒¯𝑢superscript𝑢italic-ϑdifferential-dsuperscript𝑢\displaystyle+c\Omega_{0}^{2}(u_{0})\delta\mathscr{F}\mathcal{A}\int_{u_{0}}^{u}|u^{\prime}|^{-2}(1+|\psi|)\cdot(|u^{\prime}||L\phi|+|u^{\prime}||\Omega\widehat{\chi}|)(\underline{u},u^{\prime},\vartheta)\mathrm{d}u^{\prime}
+cΩ02(u0)u0u|u|1||u|L¯ϕ(u¯,u,ϑ)ψ(u)||u||Lϕ|(u¯,u,ϑ)du.𝑐superscriptsubscriptΩ02subscript𝑢0superscriptsubscriptsubscript𝑢0𝑢superscriptsuperscript𝑢1superscript𝑢¯𝐿italic-ϕ¯𝑢superscript𝑢italic-ϑ𝜓superscript𝑢superscript𝑢𝐿italic-ϕ¯𝑢superscript𝑢italic-ϑdifferential-dsuperscript𝑢\displaystyle+c\Omega_{0}^{2}(u_{0})\int_{u_{0}}^{u}|u^{\prime}|^{-1}||u^{\prime}|\underline{L}\phi(\underline{u},u^{\prime},\vartheta)-\psi(u^{\prime})|\cdot|u^{\prime}||L\phi|(\underline{u},u^{\prime},\vartheta)\mathrm{d}u^{\prime}.

Now integrate the above inequality over u¯¯𝑢\underline{u}, and assume, for all u[u0,u1]𝑢subscript𝑢0subscript𝑢1u\in[u_{0},u_{1}],

0δ[|u|2(|Ωχ^|2+2|Lϕ|2)](u¯,u,ϑ)du¯20δ[|u|2(|Ωχ^|2+2|Lϕ|2)](u¯,u0,ϑ)du¯2I0(ϑ),superscriptsubscript0𝛿delimited-[]superscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢𝑢italic-ϑdifferential-dsuperscript¯𝑢2superscriptsubscript0𝛿delimited-[]superscriptsuperscript𝑢2superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢0italic-ϑdifferential-dsuperscript¯𝑢2subscript𝐼0italic-ϑ\displaystyle\int_{0}^{\delta}[|u|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})](\underline{u}^{\prime},u,\vartheta)\mathrm{d}\underline{u}^{\prime}\leq 2\int_{0}^{\delta}[|u^{\prime}|^{2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})](\underline{u}^{\prime},u_{0},\vartheta)\mathrm{d}\underline{u}^{\prime}\triangleq 2I_{0}(\vartheta), (5.17)

we have, for C1subscript𝐶1C_{1} sufficiently large, using (5.4) and (5.5),

|0δ[|u|2|Ωχ^|2+2|u|2|Lϕ|2](u¯,u,ϑ)du¯I0(ϑ)|superscriptsubscript0𝛿delimited-[]superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscript𝐿italic-ϕ2superscript¯𝑢𝑢italic-ϑdifferential-dsuperscript¯𝑢subscript𝐼0italic-ϑ\displaystyle\left|\int_{0}^{\delta}[|u|^{2}|\Omega\widehat{\chi}|^{2}+2|u|^{2}|L\phi|^{2}](\underline{u}^{\prime},u,\vartheta)\mathrm{d}\underline{u}^{\prime}-I_{0}(\vartheta)\right|
\displaystyle\leq cC1(I0(ϑ)+Ω02(u0)δ12I012(ϑ))+4δ12I012(ϑ)u0uΩ02h|ψ||u|du12I0(ϑ).𝑐superscript𝐶1subscript𝐼0italic-ϑsuperscriptsubscriptΩ02subscript𝑢0superscript𝛿12superscriptsubscript𝐼012italic-ϑ4superscript𝛿12superscriptsubscript𝐼012italic-ϑsuperscriptsubscriptsubscript𝑢0𝑢superscriptsubscriptΩ02𝜓superscript𝑢differential-dsuperscript𝑢12subscript𝐼0italic-ϑ\displaystyle cC^{-1}\cdot(I_{0}(\vartheta)+\Omega_{0}^{2}(u_{0})\delta^{\frac{1}{2}}I_{0}^{\frac{1}{2}}(\vartheta))+4\delta^{\frac{1}{2}}I_{0}^{\frac{1}{2}}(\vartheta)\int_{u_{0}}^{u}\frac{\Omega_{0}^{2}h|\psi|}{|u^{\prime}|}\mathrm{d}u^{\prime}\leq\frac{1}{2}I_{0}(\vartheta).

This improves (5.17) and therefore holds without assuming (5.17). On the other hand, we derive that for any u[u0,u1]𝑢subscript𝑢0subscript𝑢1u\in[u_{0},u_{1}],

0δ[|u|2|Ωχ^|2+2|u|2|Lϕ|2](u¯,u,ϑ)du¯12I0(ϑ)superscriptsubscript0𝛿delimited-[]superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscript𝐿italic-ϕ2superscript¯𝑢𝑢italic-ϑdifferential-dsuperscript¯𝑢12subscript𝐼0italic-ϑ\displaystyle\int_{0}^{\delta}[|u|^{2}|\Omega\widehat{\chi}|^{2}+2|u|^{2}|L\phi|^{2}](\underline{u}^{\prime},u,\vartheta)\mathrm{d}\underline{u}^{\prime}\geq\frac{1}{2}I_{0}(\vartheta)

From this inequality, (5.4) and (5.3), we have, in particular,

0δ[|u1|2|Ωχ^|2+2|u1|2|Lϕ|2](u¯,u1,ϑ)du¯172Ω02(u1)|u1|.superscriptsubscript0𝛿delimited-[]superscriptsubscript𝑢12superscriptΩ^𝜒22superscriptsubscript𝑢12superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢1italic-ϑdifferential-dsuperscript¯𝑢172superscriptsubscriptΩ02subscript𝑢1subscript𝑢1\displaystyle\int_{0}^{\delta}[|u_{1}|^{2}|\Omega\widehat{\chi}|^{2}+2|u_{1}|^{2}|L\phi|^{2}](\underline{u}^{\prime},u_{1},\vartheta)\mathrm{d}\underline{u}^{\prime}\geq\frac{17}{2}\Omega_{0}^{2}(u_{1})|u_{1}|. (5.18)

Now we integrate again the equation (4.25) along u¯¯𝑢\underline{u} curve at u=u1𝑢subscript𝑢1u=u_{1}, using the estimate (4.8), we have, for C1subscript𝐶1C_{1} sufficiently large,

trχ(δ,u1,ϑ)2h(u1)|u1|trsuperscript𝜒𝛿subscript𝑢1italic-ϑ2subscript𝑢1subscript𝑢1absent\displaystyle\mathrm{tr}\chi^{\prime}(\delta,u_{1},\vartheta)-\frac{2h(u_{1})}{|u_{1}|}\leq 0δΩ2(|Ωχ^|2+2|Lϕ|2)(u¯,u1,ϑ)du¯superscriptsubscript0𝛿superscriptΩ2superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢1italic-ϑdifferential-dsuperscript¯𝑢\displaystyle-\int_{0}^{\delta}\Omega^{-2}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u}^{\prime},u_{1},\vartheta)\mathrm{d}\underline{u}^{\prime}
\displaystyle\leq 14Ω02(u1)0δ(|Ωχ^|2+2|Lϕ|2)(u¯,u1,ϑ)du¯14superscriptsubscriptΩ02subscript𝑢1superscriptsubscript0𝛿superscriptΩ^𝜒22superscript𝐿italic-ϕ2superscript¯𝑢subscript𝑢1italic-ϑdifferential-dsuperscript¯𝑢\displaystyle-\frac{1}{4}\Omega_{0}^{-2}(u_{1})\int_{0}^{\delta}(|\Omega\widehat{\chi}|^{2}+2|L\phi|^{2})(\underline{u}^{\prime},u_{1},\vartheta)\mathrm{d}\underline{u}^{\prime}
\displaystyle\leq 17162|u1|17162subscript𝑢1\displaystyle-\frac{17}{16}\cdot\frac{2}{|u_{1}|}

which implies trχ(δ,u1,ϑ)<0trsuperscript𝜒𝛿subscript𝑢1italic-ϑ0\mathrm{tr}\chi^{\prime}(\delta,u_{1},\vartheta)<0.

 

As a special case, we have

Theorem 5.2.

Consider the characteristic initial value problem as in Theorem 3.1. Suppose that the initial data on C¯0subscript¯𝐶0\underline{C}_{0} is spherically symmetric with Ω0(u0)1subscriptΩ0subscript𝑢01\Omega_{0}(u_{0})\leq 1 and the data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} obeys

max{supu0uu1|φ(u)|,|u0|sup0u¯δ(Ωχ^7(u¯,u0)+ω,Lϕ5(u¯,u0))}Ω02(u0)a|log|u1||u0||\max\left\{\sup_{u_{0}\leq u\leq u_{1}}|\varphi(u)|,|u_{0}|\sup_{0\leq\underline{u}\leq\delta}\left(\|\Omega\widehat{\chi}\|_{\mathbb{H}^{7}(\underline{u},u_{0})}+\|\omega,L\phi\|_{\mathbb{H}^{5}(\underline{u},u_{0})}\right)\right\}\leq\Omega_{0}^{2}(u_{0})a\left|\log\frac{|u_{1}|}{|u_{0}|}\right|

for some a1𝑎1a\geq 1. The proof is then completed after we note that

supu~uu~(~1|φ(u)|)0subscriptsupremum~𝑢superscript𝑢subscript~𝑢subscriptsuperscript~1𝜑superscript𝑢0\displaystyle\sup_{\widetilde{u}\leq u^{\prime}\leq\widetilde{u}_{*}}(\widetilde{\mathscr{F}}^{-1}_{*}|\varphi(u^{\prime})|)\to 0

as u~0~𝑢0\widetilde{u}\to 0 (because of (LABEL:def-F*utilde) and (LABEL:F*>=1)).

The final proposition we will prove is

Proposition 5.2.

There exists some ε4subscript𝜀4\varepsilon_{4} such that if |u~|<ε4~𝑢subscript𝜀4|\widetilde{u}|<\varepsilon_{4} and

Ω0γ2(u~)f(u~)2,superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢2\displaystyle\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u})\geq 2,

then we have

infϑS20δ~(|u~|2|Ωχ^|2+2|u~|2|Lϕ|2)(u¯,u~,ϑ)du¯17C2Ω~4δ~|log|u~||u~||.subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0~𝛿superscript~𝑢2superscriptΩ^𝜒22superscript~𝑢2superscript𝐿italic-ϕ2superscript¯𝑢~𝑢italic-ϑdifferential-dsuperscript¯𝑢17superscript𝐶2superscript~Ω4~𝛿subscript~𝑢~𝑢\displaystyle\inf_{\vartheta\in S^{2}}\int_{0}^{\widetilde{\delta}}(|\widetilde{u}|^{2}|\Omega\widehat{\chi}|^{2}+2|\widetilde{u}|^{2}|L\phi|^{2})(\underline{u}^{\prime},\widetilde{u},\vartheta)\mathrm{d}\underline{u}^{\prime}\geq 17C^{2}\widetilde{\Omega}^{4}\widetilde{\delta}\left|\log\frac{|\widetilde{u}_{*}|}{|\widetilde{u}|}\right|. (5.19)
Proof.

We integrate the equation (4.14) from u=u0𝑢subscript𝑢0u=u_{0} to u=u~𝑢~𝑢u=\widetilde{u}, and obtain, similar to (LABEL:estimate-uLphi-varphi-improve), for all 0u¯δ~0¯𝑢~𝛿0\leq\underline{u}\leq\widetilde{\delta}, ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2},

|(|u~|Lϕ(u¯,u~,ϑ)φ(u))(|u0|Lϕ(u¯,u0,ϑ)φ(u0))|δ~|u~|1~2𝒲~𝒜~2,less-than-or-similar-to~𝑢𝐿italic-ϕ¯𝑢~𝑢italic-ϑ𝜑𝑢subscript𝑢0𝐿italic-ϕ¯𝑢subscript𝑢0italic-ϑ𝜑subscript𝑢0~𝛿superscript~𝑢1superscript~2~𝒲superscript~𝒜2\displaystyle|(|\widetilde{u}|L\phi(\underline{u},\widetilde{u},\vartheta)-\varphi(u))-(|u_{0}|L\phi(\underline{u},u_{0},\vartheta)-\varphi(u_{0}))|\lesssim\widetilde{\delta}|\widetilde{u}|^{-1}\widetilde{\mathscr{F}}^{2}\widetilde{\mathscr{W}}\widetilde{\mathcal{A}}^{2},

which implies

||u~|Lϕ(u¯,u~,ϑ)|||u0|Lϕ(u¯,u0,ϑ)+(φ(u~)φ(u0))|cδ~|u~|1~2𝒲~𝒜~2,~𝑢𝐿italic-ϕ¯𝑢~𝑢italic-ϑsuperscriptsubscript𝑢0𝐿italic-ϕ¯𝑢subscript𝑢0italic-ϑ𝜑~𝑢𝜑subscript𝑢0𝑐~𝛿~𝑢1superscript~2~𝒲superscript~𝒜2\displaystyle||\widetilde{u}|L\phi(\underline{u},\widetilde{u},\vartheta)|\geq||u_{0}|L\phi(\underline{u},u_{0},\vartheta)+(\varphi(\widetilde{u})-\varphi(u_{0}))|-c\widetilde{\delta}|\widetilde{u}|^{-1}\widetilde{\mathscr{F}}^{2}\widetilde{\mathscr{W}}\widetilde{\mathcal{A}}^{2}, (5.20)

On the other hand, consider the equation (4.23), written in the following form:

u(|u|2|Ωχ^|2)=2|u|2Ω2(/ ^η+η^η+/ ϕ^/ ϕ/ b(Ωχ^)),Ωχ^(Ωtrχ¯+2|u|)|u|2|Ωχ^|2Ωtrχ|u|2Ωχ¯^,Ωχ^,𝑢superscript𝑢2superscriptΩ^𝜒22superscript𝑢2superscriptΩ2/ ^tensor-product𝜂𝜂^tensor-product𝜂/ italic-ϕ^tensor-product/ italic-ϕsubscript/ 𝑏Ω^𝜒Ω^𝜒Ωtr¯𝜒2𝑢superscript𝑢2superscriptΩ^𝜒2Ωtr𝜒superscript𝑢2Ω^¯𝜒Ω^𝜒\begin{split}\frac{\partial}{\partial u}(|u|^{2}|\Omega\widehat{\chi}|^{2})=&2|u|^{2}\langle\Omega^{2}(\mbox{$\nabla\mkern-13.0mu/$ }\widehat{\otimes}\eta+\eta\widehat{\otimes}\eta+\mbox{$\nabla\mkern-13.0mu/$ }\phi\widehat{\otimes}\mbox{$\nabla\mkern-13.0mu/$ }\phi-\mbox{$\nabla\mkern-13.0mu/$ }_{b}(\Omega\widehat{\chi})),\Omega\widehat{\chi}\rangle\\ &-\left(\Omega\mathrm{tr}\underline{\chi}+\frac{2}{|u|}\right)|u|^{2}|\Omega\widehat{\chi}|^{2}-\Omega\mathrm{tr}\chi|u|^{2}\langle\Omega\widehat{\underline{\chi}},\Omega\widehat{\chi}\rangle,\end{split}

we will have, for all 0u¯δ~0¯𝑢~𝛿0\leq\underline{u}\leq\widetilde{\delta}, ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2}

|u~|2|Ωχ^(u¯,u~,ϑ)|2|u0|2|Ωχ^(u¯,u0,ϑ)|2cδ~|u~|1~3𝒜~3.superscript~𝑢2superscriptΩ^𝜒¯𝑢~𝑢italic-ϑ2superscriptsubscript𝑢02superscriptΩ^𝜒¯𝑢subscript𝑢0italic-ϑ2𝑐~𝛿superscript~𝑢1superscript~3superscript~𝒜3\displaystyle|\widetilde{u}|^{2}|\Omega\widehat{\chi}(\underline{u},\widetilde{u},\vartheta)|^{2}\geq|u_{0}|^{2}|\Omega\widehat{\chi}(\underline{u},u_{0},\vartheta)|^{2}-c\widetilde{\delta}|\widetilde{u}|^{-1}\widetilde{\mathscr{F}}^{3}\widetilde{\mathcal{A}}^{3}. (5.21)

Integrating the sqaure of (5.20) and (5.21) over u¯¯𝑢\underline{u}, together with (LABEL:estimate-deltau-1tilde), we have

1δ~0δ~(|u~|2|Ωχ^|2+2|u~|2|Lϕ|2)(u¯,u~,ϑ)du¯1~𝛿superscriptsubscript0~𝛿superscript~𝑢2superscriptΩ^𝜒22superscript~𝑢2superscript𝐿italic-ϕ2superscript¯𝑢~𝑢italic-ϑdifferential-dsuperscript¯𝑢\displaystyle\frac{1}{\widetilde{\delta}}\int_{0}^{\widetilde{\delta}}(|\widetilde{u}|^{2}|\Omega\widehat{\chi}|^{2}+2|\widetilde{u}|^{2}|L\phi|^{2})(\underline{u}^{\prime},\widetilde{u},\vartheta)\mathrm{d}\underline{u}^{\prime}
\displaystyle\geq 1δ~0δ~(|u0|2|Ωχ^(u¯,u0,ϑ)|2+||u0|Lϕ(u¯,u0,ϑ)+(φ(u~)φ(u0))|2)du¯cC2e117C2Ω~2γF31~𝛿superscriptsubscript0~𝛿superscriptsubscript𝑢02superscriptΩ^𝜒¯𝑢subscript𝑢0italic-ϑ2superscriptsubscript𝑢0𝐿italic-ϕ¯𝑢subscript𝑢0italic-ϑ𝜑~𝑢𝜑subscript𝑢02differential-dsuperscript¯𝑢𝑐superscript𝐶2superscripte117superscript𝐶2superscript~Ω2𝛾superscript𝐹3\displaystyle\frac{1}{\widetilde{\delta}}\int_{0}^{\widetilde{\delta}}(|u_{0}|^{2}|\Omega\widehat{\chi}(\underline{u},u_{0},\vartheta)|^{2}+||u_{0}|L\phi(\underline{u},u_{0},\vartheta)+(\varphi(\widetilde{u})-\varphi(u_{0}))|^{2})\mathrm{d}\underline{u}^{\prime}-cC^{-2}\mathrm{e}^{-\frac{1}{17}C^{-2}\widetilde{\Omega}^{-2-\gamma}}F^{3}
=\displaystyle= f(u~)cC2e117C2Ω~2γF3.𝑓~𝑢𝑐superscript𝐶2superscripte117superscript𝐶2superscript~Ω2𝛾superscript𝐹3\displaystyle f(\widetilde{u})-cC^{-2}\mathrm{e}^{-\frac{1}{17}C^{-2}\widetilde{\Omega}^{-2-\gamma}}F^{3}.

Then there exists some ε4subscript𝜀4\varepsilon_{4} such that if |u~|<ε4~𝑢subscript𝜀4|\widetilde{u}|<\varepsilon_{4}, such that

cC2e118C2Ω~2γF312Ω~2γ.𝑐superscript𝐶2superscripte118superscript𝐶2superscript~Ω2𝛾superscript𝐹312superscript~Ω2𝛾\displaystyle cC^{-2}\mathrm{e}^{-\frac{1}{18}C^{-2}\widetilde{\Omega}^{-2-\gamma}}F^{3}\leq\frac{1}{2}\widetilde{\Omega}^{2-\gamma}.

If Ω0γ2(u~)f(u~;γ)2superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢𝛾2\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u};\gamma)\geq 2, we then have

1δ~0δ~(|u~|2|Ωχ^|2+2|u~|2|Lϕ|2)(u¯,u~,ϑ)du¯1~𝛿superscriptsubscript0~𝛿superscript~𝑢2superscriptΩ^𝜒22superscript~𝑢2superscript𝐿italic-ϕ2superscript¯𝑢~𝑢italic-ϑdifferential-dsuperscript¯𝑢absent\displaystyle\frac{1}{\widetilde{\delta}}\int_{0}^{\widetilde{\delta}}(|\widetilde{u}|^{2}|\Omega\widehat{\chi}|^{2}+2|\widetilde{u}|^{2}|L\phi|^{2})(\underline{u}^{\prime},\widetilde{u},\vartheta)\mathrm{d}\underline{u}^{\prime}\geq 2Ω~2γ12Ω~2γΩ~2γ.2superscript~Ω2𝛾12superscript~Ω2𝛾superscript~Ω2𝛾\displaystyle 2\widetilde{\Omega}^{2-\gamma}-\frac{1}{2}\widetilde{\Omega}^{2-\gamma}\geq\widetilde{\Omega}^{2-\gamma}.

In view of (LABEL:u*tilde) and (LABEL:estimate-Atilde*),

1δ~0δ~(|u~|2|Ωχ^|2+2|u~|2|Lϕ|2)(u¯,u~,ϑ)du¯17C2Ω~4|log|u~||u~||,1~𝛿superscriptsubscript0~𝛿superscript~𝑢2superscriptΩ^𝜒22superscript~𝑢2superscript𝐿italic-ϕ2superscript¯𝑢~𝑢italic-ϑdifferential-dsuperscript¯𝑢17superscript𝐶2superscript~Ω4subscript~𝑢~𝑢\displaystyle\frac{1}{\widetilde{\delta}}\int_{0}^{\widetilde{\delta}}(|\widetilde{u}|^{2}|\Omega\widehat{\chi}|^{2}+2|\widetilde{u}|^{2}|L\phi|^{2})(\underline{u}^{\prime},\widetilde{u},\vartheta)\mathrm{d}\underline{u}^{\prime}\geq 17C^{2}\widetilde{\Omega}^{4}\left|\log\frac{|\widetilde{u}_{*}|}{|\widetilde{u}|}\right|,

for all ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2} which implies (5.19).  

By (LABEL:deltatilde), (LABEL:def-F*utilde), (LABEL:estimate-Atilde*) and (5.19), we can apply Theorem 5.2 with a=1𝑎1a=1, and conclude that Sδ~,u~subscript𝑆~𝛿subscript~𝑢S_{\widetilde{\delta},\widetilde{u}_{*}} is a closed trapped surface if |u~|<ε0:=min{ε1,ε2,ε3,ε4}~𝑢subscript𝜀0assignsubscript𝜀1subscript𝜀2subscript𝜀3subscript𝜀4|\widetilde{u}|<\varepsilon_{0}:=\min\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\}. This completes the proof of Case 2.

Remark 5.1.

The additional bound (LABEL:conditionE), or the stronger bound (LABEL:conditionE'), which is more natural, are introduced to guarantee the validity of the first one of the smallness assumptions (3.4) in both cases. We will use instead the bound (LABEL:conditionE) in the actual proof in the next section. On the other hand, if we assume the rate of Ω0subscriptΩ0\Omega_{0} tending to zero is sufficiently fast, for example,

supu0u<0e117C2Ω02(u)|log|u||u0||<+,subscriptsupremumsubscript𝑢0𝑢0superscripte117superscript𝐶2superscriptsubscriptΩ02𝑢𝑢subscript𝑢0\displaystyle\sup_{u_{0}\leq u<0}\mathrm{e}^{-\frac{1}{17}C^{-2}\Omega_{0}^{-2}(u)}\left|\log\frac{|u|}{|u_{0}|}\right|<+\infty,

which is clearly not optimal, then we can see from the proof that we do not need the bound (LABEL:conditionE).

6. Gravitational perturbations

Recalling that the initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} consists the conformal metric g/^\widehat{\mbox{$g\mkern-9.0mu/$}}, the lapse ΩΩ\Omega and the scalar field function ϕitalic-ϕ\phi. In this section, we should consider non-smooth initial data.

6.1. The instability theorems for non-smooth initial data

First of all, we state and prove the instability theorem for non-smooth initial data.

Theorem 6.1.

Suppose that the initial data on C¯0subscript¯𝐶0\underline{C}_{0} is smooth, spherically symmetric, singular at the vertex, and φ(u)𝜑𝑢\varphi(u) is bounded. The data on Cu0subscript𝐶subscript𝑢0C_{u_{0}} is given on a fixed intervel, say u¯[0,1]¯𝑢01\underline{u}\in[0,1], not necessarily smooth, satisfies

|u0|supu¯[0,1](Ωχ^10(u¯,u0)+ω8(u¯,u0)+Lϕ8(u¯,u0))Asubscript𝑢0subscriptsupremum¯𝑢01subscriptdelimited-∥∥Ω^𝜒superscript10¯𝑢subscript𝑢0subscriptdelimited-∥∥𝜔superscript8¯𝑢subscript𝑢0subscriptdelimited-∥∥𝐿italic-ϕsuperscript8¯𝑢subscript𝑢0𝐴\begin{split}|u_{0}|\sup_{\underline{u}\in[0,1]}\left(\|\Omega\widehat{\chi}\|_{\mathbb{H}^{10}(\underline{u},u_{0})}+\|\omega\|_{\mathbb{H}^{8}(\underline{u},u_{0})}+\|L\phi\|_{\mathbb{H}^{8}(\underline{u},u_{0})}\right)\leq A\end{split} (6.1)

and

min{supδ(0,1]|u0|(|u0|/ )Lϕ𝕃[0,δ]27(u0)2|logC2δ|u0||,supu0u<0e117C2Ω02(u)|log|u||u0||}Esubscriptsupremum𝛿01evaluated-atsubscript𝑢0subscript𝑢0/ 𝐿italic-ϕsubscriptsuperscript𝕃20𝛿superscript7subscript𝑢02superscript𝐶2𝛿subscript𝑢0subscriptsupremumsubscript𝑢0𝑢0superscripte117superscript𝐶2superscriptsubscriptΩ02𝑢𝑢subscript𝑢0𝐸\begin{split}\min\left\{\sup_{\delta\in(0,1]}\||u_{0}|(|u_{0}|\mbox{$\nabla\mkern-13.0mu/$ })L\phi\|^{2}_{\mathbb{L}^{2}_{[0,\delta]}\mathbb{H}^{7}(u_{0})}\left|\log\frac{C^{2}\delta}{|u_{0}|}\right|,\sup_{u_{0}\leq u<0}\mathrm{e}^{-\frac{1}{17}C^{-2}\Omega_{0}^{-2}(u)}\left|\log\frac{|u|}{|u_{0}|}\right|\right\}\leq E\end{split} (6.2)

for some A1,E𝐴1𝐸A\geq 1,E and Cmax{C1,4}𝐶subscript𝐶14C\geq\max\{C_{1},4\}. Suppose also that for some γ(0,2)𝛾02\gamma\in(0,2),

Ω0γ2(u~)f(u~;γ)32,superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢𝛾32\displaystyle\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u};\gamma)\geq 32, (6.3)

for some |u~|<ε0~𝑢subscript𝜀0|\widetilde{u}|<\varepsilon_{0} where f(u~;γ)𝑓~𝑢𝛾f(\widetilde{u};\gamma) is the function defined in (LABEL:def-f) and ε0subscript𝜀0\varepsilon_{0} is given in Theorem LABEL:instabilitytheorem.

Then there exists a pair of numbers (δ~,u~)~𝛿subscript~𝑢(\widetilde{\delta},\widetilde{u}_{*}) such that the following conclusions hold: Let (g/^n,Ωn,Lϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},L\phi_{n}) be a sequence of smooth initial data on Cu0subscript𝐶subscript𝑢0C_{u_{0}}, satisfying (6.1) and (6.2) with the same constants A𝐴A and E𝐸E. Suppose that

12|ξ|g/|ξ|g/n2|ξ|g/\displaystyle\frac{1}{2}|\xi|_{\mbox{$g\mkern-9.0mu/$}}\leq|\xi|_{\mbox{$g\mkern-9.0mu/$}_{n}}\leq 2|\xi|_{\mbox{$g\mkern-9.0mu/$}}

for all 222-covariant tensor field ξ𝜉\xi and all n𝑛n, and

limnsupϑS201|u0|2|Ωnχ^nΩχ^|g/2(u¯,u0,ϑ)du¯=0,\displaystyle\lim_{n\to\infty}\sup_{\vartheta\in S^{2}}\int_{0}^{1}|u_{0}|^{2}|\Omega_{n}\widehat{\chi}_{n}-\Omega\widehat{\chi}|_{\mbox{$g\mkern-9.0mu/$}}^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}=0, (6.4)
limnsupϑS201|u0|2|Lϕn(u¯,u0,ϑ)Lϕ(u¯,u0,ϑ)|2du¯=0.subscript𝑛subscriptsupremumitalic-ϑsuperscript𝑆2superscriptsubscript01superscriptsubscript𝑢02superscript𝐿subscriptitalic-ϕ𝑛¯𝑢subscript𝑢0italic-ϑ𝐿italic-ϕ¯𝑢subscript𝑢0italic-ϑ2differential-d¯𝑢0\displaystyle\lim_{n\to\infty}\sup_{\vartheta\in S^{2}}\int_{0}^{1}|u_{0}|^{2}|L\phi_{n}(\underline{u},u_{0},\vartheta)-L\phi(\underline{u},u_{0},\vartheta)|^{2}\mathrm{d}\underline{u}=0. (6.5)

Then there exists some N𝑁N such that for all n>N𝑛𝑁n>N, the sphere Sδ~,u~subscript𝑆~𝛿subscript~𝑢S_{\widetilde{\delta},\widetilde{u}_{*}} is a closed trapped surface in the maximal development of the initial data (g/^n,Ωn,Lϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},L\phi_{n}). Moreover, the sphere Sδ~,u~subscript𝑆~𝛿subscript~𝑢S_{\widetilde{\delta},\widetilde{u}_{*}} is uniformly strictly trapped, in the sense that for all n>N𝑛𝑁n>N,

(trχ)n|Sδ~,u~17162|u~|,Ωntrχ¯n|Sδ~,u~1|u~|.formulae-sequenceevaluated-atsubscripttrsuperscript𝜒𝑛subscript𝑆~𝛿subscript~𝑢17162subscript~𝑢evaluated-atsubscriptΩ𝑛trsubscript¯𝜒𝑛subscript𝑆~𝛿~𝑢1subscript~𝑢\displaystyle(\mathrm{tr}\chi^{\prime})_{n}|_{S_{\widetilde{\delta},\widetilde{u}_{*}}}\leq-\frac{17}{16}\cdot\frac{2}{|\widetilde{u}_{*}|},\ \ \Omega_{n}\mathrm{tr}\underline{\chi}_{n}|_{S_{\widetilde{\delta},\widetilde{u}}}\leq-\frac{1}{|\widetilde{u}_{*}|}.
Proof.

Choose numbers δ~~𝛿\widetilde{\delta} and u~subscript~𝑢\widetilde{u}_{*} defined through (LABEL:u*tilde) and (LABEL:deltatilde) in terms of u~~𝑢\widetilde{u}. The trapped surface being strictly trapped uniformly is followed directly from the proof of Theorem 5.1. We only need to verify that for all n>N𝑛𝑁n>N for some N𝑁N, (LABEL:instabilitycondition) holds and then apply Theorem LABEL:instabilitytheorem and Remark 5.1. Indeed, from the assumptions, we can find some N1subscript𝑁1N_{1} such that for all n>N1𝑛subscript𝑁1n>N_{1},

supϑS20δ~(u~;γ)|u0|2|Ωnχ^nΩχ^|g/2(u¯,u0,ϑ)du¯18δ~(u~;γ)f(u~;γ),\displaystyle\sup_{\vartheta\in S^{2}}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}|u_{0}|^{2}|\Omega_{n}\widehat{\chi}_{n}-\Omega\widehat{\chi}|_{\mbox{$g\mkern-9.0mu/$}}^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}\leq\frac{1}{8}\widetilde{\delta}(\widetilde{u};\gamma)f(\widetilde{u};\gamma),

then we obtain

1δ~(u~;γ)0δ~(u~;γ)|u0|2|Ωnχ^n|g/n2(u¯,u0,ϑ)du¯\displaystyle\frac{1}{\widetilde{\delta}(\widetilde{u};\gamma)}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}|u_{0}|^{2}|\Omega_{n}\widehat{\chi}_{n}|_{\mbox{$g\mkern-9.0mu/$}_{n}}^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}
\displaystyle\geq 14δ~(u~;γ)(120δ~(u~;γ)|u0|2|Ωχ^|g/2(u¯,u0,ϑ)du¯0δ~(u~;γ)|u0|2|Ωnχ^nΩχ^|g/2(u¯,u0,ϑ)du¯)\displaystyle\frac{1}{4\widetilde{\delta}(\widetilde{u};\gamma)}\left(\frac{1}{2}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}|u_{0}|^{2}|\Omega\widehat{\chi}|_{\mbox{$g\mkern-9.0mu/$}}^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}-\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}|u_{0}|^{2}|\Omega_{n}\widehat{\chi}_{n}-\Omega\widehat{\chi}|_{\mbox{$g\mkern-9.0mu/$}}^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}\right)
\displaystyle\geq 18δ~(u~;γ)0δ~(u~;γ)|u0|2|Ωχ^|g/2(u¯,u0,ϑ)du¯132δ~(u~;γ)f(u~;γ).\displaystyle\frac{1}{8\widetilde{\delta}(\widetilde{u};\gamma)}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}|u_{0}|^{2}|\Omega\widehat{\chi}|_{\mbox{$g\mkern-9.0mu/$}}^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}-\frac{1}{32}\widetilde{\delta}(\widetilde{u};\gamma)f(\widetilde{u};\gamma).

Similar argument allows us to find some N2subscript𝑁2N_{2} such that for n>N2𝑛subscript𝑁2n>N_{2},

1δ~(u~;γ)0δ~(u~;γ)||u0|Lϕn(u¯,u0,ϑ)+(φ(u~)φ(u0)|2du¯\displaystyle\frac{1}{\widetilde{\delta}(\widetilde{u};\gamma)}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}||u_{0}|L\phi_{n}(\underline{u},u_{0},\vartheta)+(\varphi(\widetilde{u})-\varphi(u_{0})|^{2}\mathrm{d}\underline{u}
\displaystyle\geq 18δ~(u~;γ)0δ~(u~;γ)||u0|Lϕ(u¯,u0,ϑ)+(φ(u~)φ(u0)|2du¯132δ~(u~;γ)f(u~;γ).\displaystyle\frac{1}{8\widetilde{\delta}(\widetilde{u};\gamma)}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}||u_{0}|L\phi(\underline{u},u_{0},\vartheta)+(\varphi(\widetilde{u})-\varphi(u_{0})|^{2}\mathrm{d}\underline{u}-\frac{1}{32}\widetilde{\delta}(\widetilde{u};\gamma)f(\widetilde{u};\gamma).

Summing up the above two inequalities implies that (LABEL:instabilitycondition) holds for (g/^n,Ωn,Lϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},L\phi_{n}) if n>N=max{N1,N2}𝑛𝑁subscript𝑁1subscript𝑁2n>N=\max\{N_{1},N_{2}\} from the assumption (6.3).  

From the conclusions of the above theorem, if the solutions of the Einstein equations with initial data (g/^n,Ωn,ϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},\phi_{n}) converge, in any senses such that (trχ)nsubscripttrsuperscript𝜒𝑛(\mathrm{tr}\chi^{\prime})_{n} and Ωntrχ¯nsubscriptΩ𝑛trsubscript¯𝜒𝑛\Omega_{n}\mathrm{tr}\underline{\chi}_{n} converge pointwisely on Sδ~,u~subscript𝑆~𝛿~𝑢S_{\widetilde{\delta},\widetilde{u}}, then the limiting spacetime has a closed trapped surface. Therefore we may say, that the future development of such initial data has a closed trapped surface.

Finally, we have the following.

Theorem 6.2.

Suppose that the initial data on C¯0subscript¯𝐶0\underline{C}_{0} is smooth, spherically symmetric and singular at the vertex. The data (g/^,Ω,ϕ)(\widehat{\mbox{$g\mkern-9.0mu/$}},\Omega,\phi) on Cu0subscript𝐶subscript𝑢0C_{u_{0}} satisfies (6.1) and (6.2) for some Cmax{C1,4}𝐶subscript𝐶14C\geq\max\{C_{1},4\}. If φ(u)𝜑𝑢\varphi(u) is bounded, we assume in addition

lim supu~0Ω0γ2(u~)f(u~;γ)>32subscriptlimit-supremum~𝑢superscript0superscriptsubscriptΩ0𝛾2~𝑢𝑓~𝑢𝛾32\displaystyle\limsup_{\widetilde{u}\to 0^{-}}\Omega_{0}^{\gamma-2}(\widetilde{u})f(\widetilde{u};\gamma)>32 (6.6)

for some γ(0,2)𝛾02\gamma\in(0,2).

Then we can find two sequences δ~k0+subscript~𝛿𝑘superscript0\widetilde{\delta}_{k}\to 0^{+} and u~1,k0subscript~𝑢1𝑘superscript0\widetilde{u}_{1,k}\to 0^{-}, such that the following conclusions hold: Let (g/^n,Ωn,ϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},\phi_{n}) be a sequence of smooth initial data satisfying the same assumptions as in the statement of Theorem 6.1. Then for every k𝑘k, there exists some N=Nk𝑁subscript𝑁𝑘N=N_{k}, such that for all n>Nk𝑛subscript𝑁𝑘n>N_{k},

(trχ)n|Sδ~k,u~1,k17162|u~1,k|,Ωntrχ¯n|Sδ~k,u~1,k1|u~1,k|.formulae-sequenceevaluated-atsubscripttrsuperscript𝜒𝑛subscript𝑆subscript~𝛿𝑘subscript~𝑢1𝑘17162subscript~𝑢1𝑘evaluated-atsubscriptΩ𝑛trsubscript¯𝜒𝑛subscript𝑆subscript~𝛿𝑘subscript~𝑢1𝑘1subscript~𝑢1𝑘\displaystyle(\mathrm{tr}\chi^{\prime})_{n}|_{S_{\widetilde{\delta}_{k},\widetilde{u}_{1,k}}}\leq-\frac{17}{16}\cdot\frac{2}{|\widetilde{u}_{1,k}|},\ \ \Omega_{n}\mathrm{tr}\underline{\chi}_{n}|_{S_{\widetilde{\delta}_{k},\widetilde{u}_{1,k}}}\leq-\frac{1}{|\widetilde{u}_{1,k}|}.

In particular, Sδ~k,u~1,ksubscript𝑆subscript~𝛿𝑘subscript~𝑢1𝑘S_{\widetilde{\delta}_{k},\widetilde{u}_{1,k}} is trapped in the maximal development of (g/^n,Ωn,Lϕn)(\widehat{\mbox{$g\mkern-9.0mu/$}}_{n},\Omega_{n},L\phi_{n}) for all n>Nk𝑛subscript𝑁𝑘n>N_{k}.

Proof.

When φ(u)𝜑𝑢\varphi(u) is bounded, this is a direct corollary of Theorem 6.1. When φ(u)𝜑𝑢\varphi(u) is unbounded, we can see from the proof of Case 1 in Theorem LABEL:instabilitytheorem that the choice of the sequence of u0,nsubscript𝑢0𝑛u_{0,n}, and therefore the locations of the closed trapped surfaces only depends on the initial data on C¯0subscript¯𝐶0\underline{C}_{0}, and the conclusion follows from this observation. Moreover, in this case, the sequences of χ^nsubscript^𝜒𝑛\widehat{\chi}_{n} and (Lϕ)nsubscript𝐿italic-ϕ𝑛(L\phi)_{n} does not need to satisfy (6.4) and (6.5).

 

Using a limiting argument, we may also say, that the future development of the such initial data has a sequence of closed trapped surfaces approaching the singularity.

Remark 6.1.

In Theorem 6.1, if the function f𝑓f in the condition (6.3) is replaced by another function fsuperscript𝑓f^{\prime} defined as

f(u~;γ)=1δ~(u~;γ)infϑS20δ~(u~;γ)|u0|2|Ωχ^|2(u¯,u0,ϑ)du¯,superscript𝑓~𝑢𝛾1~𝛿~𝑢𝛾subscriptinfimumitalic-ϑsuperscript𝑆2superscriptsubscript0~𝛿~𝑢𝛾superscriptsubscript𝑢02superscriptΩ^𝜒2¯𝑢subscript𝑢0italic-ϑdifferential-d¯𝑢\displaystyle f^{\prime}(\widetilde{u};\gamma)=\frac{1}{\widetilde{\delta}(\widetilde{u};\gamma)}\inf_{\vartheta\in S^{2}}\int_{0}^{\widetilde{\delta}(\widetilde{u};\gamma)}|u_{0}|^{2}|\Omega\widehat{\chi}|^{2}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}, (6.7)

then the converging of Lϕ𝐿italic-ϕL\phi in (6.5) is not needed.

6.2. The space of the initial data sets

The final part is to investigate the space of the initial data sets. We first choose the stereographic charts (ϑ1,ϑ2)subscriptitalic-ϑ1subscriptitalic-ϑ2(\vartheta_{1},\vartheta_{2}) (both north pole and south pole charts) on S0,u0subscript𝑆0subscript𝑢0S_{0,u_{0}} and extend them to the whole Cu0subscript𝐶subscript𝑢0C_{u_{0}} for u¯[0,1]¯𝑢01\underline{u}\in[0,1] by requiring LθA=0𝐿subscript𝜃𝐴0L\theta_{A}=0, A=1,2𝐴12A=1,2. Since we only consider the perturbations on the conformal metric g/^\widehat{\mbox{$g\mkern-9.0mu/$}} for simplicity, we will fix the lapse ΩΩ\Omega such that ΩCu¯1Hϑ8Ωsubscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐻8italic-ϑ\Omega\in C^{1}_{\underline{u}}H^{8}_{\vartheta} and the scalar field function ϕitalic-ϕ\phi such that ϕCu¯1Hϑ8italic-ϕsubscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐻8italic-ϑ\phi\in C^{1}_{\underline{u}}H^{8}_{\vartheta}. We also fix an initial data on C¯0subscript¯𝐶0\underline{C}_{0} which is smooth, spherically symmetric and singular at the vertex. Moreover, we assume that (6.2) holds for some E𝐸E where the metric on Su¯,u0subscript𝑆¯𝑢subscript𝑢0S_{\underline{u},u_{0}} is understood to be the standard round metric with radius |u0|subscript𝑢0|u_{0}| because the real metric g/g\mkern-9.0mu/ is not yet defined. If ϕitalic-ϕ\phi is assumed to be smooth, then (6.2) does hold for some E𝐸E.

In such a coordinate system, the conformal metric g/^\widehat{\mbox{$g\mkern-9.0mu/$}} can be written as a pair of symmetric positive definite matrices g/^AB\widehat{\mbox{$g\mkern-9.0mu/$}}_{AB} writing in the form

g/^AB(u¯,u0,ϑ)=|u0|2(1+14(ϑ12+ϑ22))2mAB(u¯,ϑ)\displaystyle\widehat{\mbox{$g\mkern-9.0mu/$}}_{AB}(\underline{u},u_{0},\vartheta)=\frac{|u_{0}|^{2}}{(1+\frac{1}{4}(\vartheta_{1}^{2}+\vartheta_{2}^{2}))^{2}}m_{AB}(\underline{u},\vartheta) (6.8)

where mABsubscript𝑚𝐴𝐵m_{AB} represents a pair of matrices with determinant 111 and satisfying the coordinate transformation rule, see for example Chapter 2 in [9]. For simplicity, we use a single notation ΨABsubscriptΨ𝐴𝐵\Psi_{AB} to denote the pair of matrices ΨAB,ΨABsubscriptΨ𝐴𝐵subscriptsuperscriptΨ𝐴𝐵\Psi_{AB},\Psi^{\prime}_{AB} satisfying the coordinate transformation rule. Then this pair of matrices defines a tensor ΨΨ\Psi. We then introduce the definitions of the spaces of the initial data. We remark that all the definitions, statements and proofs below depend on the initial data on C¯0subscript¯𝐶0\underline{C}_{0} we fix.

Definition 6.1.

We define \mathcal{I} to be the space of the conformal metrics g/^\widehat{\mbox{$g\mkern-9.0mu/$}} defined for u¯[0,1]¯𝑢01\underline{u}\in[0,1] and ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2} such that g/^ABCu¯1Hϑ10\widehat{\mbox{$g\mkern-9.0mu/$}}_{AB}\in C^{1}_{\underline{u}}H^{10}_{\vartheta}, and g/^(0,ϑ)\widehat{\mbox{$g\mkern-9.0mu/$}}(0,\vartheta) is the standard round metric with radius |u0|subscript𝑢0|u_{0}|, u¯g/^(0,ϑ)=0\frac{\partial}{\partial\underline{u}}\widehat{\mbox{$g\mkern-9.0mu/$}}(0,\vartheta)=0.

Definition 6.2.

We define \mathcal{E}\subset\mathcal{I} to be the collection of g/^\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I} such that the conclusion of Theorem 6.2 does not hold for any sequences δ~k0+subscript~𝛿𝑘superscript0\widetilde{\delta}_{k}\to 0^{+} and u~1,k0subscript~𝑢1𝑘superscript0\widetilde{u}_{1,k}\to 0^{-}.

Definition 6.3.

We define εsubscript𝜀\mathcal{E}_{\varepsilon}\subset\mathcal{I} to be the collection of g/^\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I} such that the conclusion of Theorem 6.1 does not hold for any (δ~,u~)~𝛿subscript~𝑢(\widetilde{\delta},\widetilde{u}_{*}) with δ~<ε~𝛿𝜀\widetilde{\delta}<\varepsilon.

Remark 6.2.

It is not difficult to use a similar argument in Chapter 2 in [9] that (6.1) holds for some A𝐴A and (6.2) holds for some different E𝐸E by comparing on Su¯,u0subscript𝑆¯𝑢subscript𝑢0S_{\underline{u},u_{0}} the real metric g/g\mkern-9.0mu/ and the standard round metric.

At last, we will prove the following precise forms of Theorem 1.4 and 1.5, the main results of the present article.

Theorem 6.3 (Precise version of Theorem 1.4).

\mathcal{E} is of first category in \mathcal{I}. This is to say, csuperscript𝑐\mathcal{E}^{c}, the complement of \mathcal{E} in \mathcal{I}, contains a subset that is a countably intersection of open and dense subsets in \mathcal{I}.

Theorem 6.4 (Precise version of Theorem 1.5).

εcsuperscriptsubscript𝜀𝑐\mathcal{E}_{\varepsilon}^{c} contains a subset that is open and dense in \mathcal{I} for all ε>0𝜀0\varepsilon>0.

Proof of Theorem 6.3.

We begin by defining 𝒩2γsubscript𝒩2𝛾\mathcal{N}_{2-\gamma}\subset\mathcal{I} such that

𝒩2γc={g/^|lim supu~0Ω0γ2(u~)f(u~;γ)>32}\displaystyle\mathcal{N}_{2-\gamma}^{c}=\{\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}|\limsup_{\widetilde{u}\to 0^{-}}\Omega_{0}^{\gamma-2}(\widetilde{u})f^{\prime}(\widetilde{u};\gamma)>32\}

where the function fsuperscript𝑓f^{\prime} is defined in (6.7). From Theorem 6.2, 𝒩2γsubscript𝒩2𝛾\mathcal{E}\subset\mathcal{N}_{2-\gamma} for any γ(0,2)𝛾02\gamma\in(0,2). Then we only need to prove that 𝒩2γsubscript𝒩2𝛾\mathcal{N}_{2-\gamma} is of first category in \mathcal{I} for some γ(0,2)𝛾02\gamma\in(0,2). We define also 𝒩2γ,εsubscript𝒩2𝛾𝜀\mathcal{N}_{2-\gamma,\varepsilon}\subset\mathcal{I} such that

𝒩2γ,εc={g/^| there exists some u~ with δ~(u~;γ)<ε such that Ω0γ2(u~)f(u~;γ)33}.\displaystyle\mathcal{N}_{2-\gamma,\varepsilon}^{c}=\{\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}|\text{ there exists some $\widetilde{u}$ with $\widetilde{\delta}(\widetilde{u};\gamma)<\varepsilon$ such that }\Omega_{0}^{\gamma-2}(\widetilde{u})f^{\prime}(\widetilde{u};\gamma)\geq 33\}.

It is clear that 𝒩2γci𝒩2γ,εicsubscript𝑖subscriptsuperscript𝒩𝑐2𝛾subscript𝜀𝑖superscriptsubscript𝒩2𝛾𝑐\mathcal{N}_{2-\gamma}^{c}\supset\bigcap_{i}\mathcal{N}^{c}_{2-\gamma,\varepsilon_{i}} for any sequences εi0subscript𝜀𝑖0\varepsilon_{i}\to 0, and therefore we only need to prove that 𝒩2γ,εcsuperscriptsubscript𝒩2𝛾𝜀𝑐\mathcal{N}_{2-\gamma,\varepsilon}^{c} is open and dense in \mathcal{I} for all ε>0𝜀0\varepsilon>0 and γ(0,2)𝛾02\gamma\in(0,2). From now on, we fix some ε>0𝜀0\varepsilon>0 and γ(0,2)𝛾02\gamma\in(0,2) arbitrarily.

Using the argument in the proof of Theorem 6.1, we know that 𝒩2γ,εcsuperscriptsubscript𝒩2𝛾𝜀𝑐\mathcal{N}_{2-\gamma,\varepsilon}^{c} is in fact open in LϑHu¯1subscriptsuperscript𝐿italic-ϑsubscriptsuperscript𝐻1¯𝑢L^{\infty}_{\vartheta}H^{1}_{\underline{u}}. Since Cu¯1Hϑ10Cu¯1Cϑ8Cϑ8Cu¯1LϑHu¯1subscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐻10italic-ϑsubscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐶8italic-ϑsubscriptsuperscript𝐶8italic-ϑsubscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐿italic-ϑsubscriptsuperscript𝐻1¯𝑢C^{1}_{\underline{u}}H^{10}_{\vartheta}\hookrightarrow C^{1}_{\underline{u}}C^{8}_{\vartheta}\hookrightarrow C^{8}_{\vartheta}C^{1}_{\underline{u}}\hookrightarrow L^{\infty}_{\vartheta}H^{1}_{\underline{u}}, then 𝒩2γ,εcsuperscriptsubscript𝒩2𝛾𝜀𝑐\mathcal{N}_{2-\gamma,\varepsilon}^{c} is also open in Cu¯1Hϑ10subscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐻10italic-ϑC^{1}_{\underline{u}}H^{10}_{\vartheta}. The denseness is followed by the following proposition.

Proposition 6.1.

Given any g/^\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}, there exists a one-parameter family g/^t\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}\in\mathcal{I} at least for sufficiently small t𝑡t such that g/^t𝒩2γ,εc\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}\in\mathcal{N}_{2-\gamma,\varepsilon}^{c} for t0𝑡0t\neq 0, and g/^tg/^\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}\to\widehat{\mbox{$g\mkern-9.0mu/$}} in Cu¯1Hϑ10subscriptsuperscript𝐶1¯𝑢subscriptsuperscript𝐻10italic-ϑC^{1}_{\underline{u}}H^{10}_{\vartheta} as t0𝑡0t\to 0.

Proof.

Fix two pairs of symmetric trace-free matrices Ψ~AB(ϑ)subscript~Ψ𝐴𝐵italic-ϑ\widetilde{\Psi}_{AB}(\vartheta) and Ψ~AB(ϑ)subscriptsuperscript~Ψ𝐴𝐵italic-ϑ\widetilde{\Psi}^{\prime}_{AB}(\vartheta) on S2=S0,u0superscript𝑆2subscript𝑆0subscript𝑢0S^{2}=S_{0,u_{0}} such that for each pS2𝑝superscript𝑆2p\in S^{2}, either Ψ~AB(p)0subscript~Ψ𝐴𝐵𝑝0\widetilde{\Psi}_{AB}(p)\neq 0 or Ψ~AB(p)0subscriptsuperscript~Ψ𝐴𝐵𝑝0\widetilde{\Psi}^{\prime}_{AB}(p)\neq 0. Then we define a pair of symmetric trace-free matrix valued functions Ψ¯AB(u¯,ϑ)subscript¯Ψ𝐴𝐵¯𝑢italic-ϑ\overline{\Psi}_{AB}(\underline{u},\vartheta) for u¯(0,1],ϑS2formulae-sequence¯𝑢01italic-ϑsuperscript𝑆2\underline{u}\in(0,1],\vartheta\in S^{2} by

Ψ¯AB(u¯,ϑ)={Ψ~AB(ϑ),24n4<u¯24n3Ψ~AB(ϑ),24n2<u¯24n1,continuously extended,otherwiseforn=0,1,2,.formulae-sequencesubscript¯Ψ𝐴𝐵¯𝑢italic-ϑcasessubscript~Ψ𝐴𝐵italic-ϑsuperscript24𝑛4¯𝑢superscript24𝑛3subscriptsuperscript~Ψ𝐴𝐵italic-ϑsuperscript24𝑛2¯𝑢superscript24𝑛1continuously extendedotherwisefor𝑛012\displaystyle\overline{\Psi}_{AB}(\underline{u},\vartheta)=\begin{cases}\widetilde{\Psi}_{AB}(\vartheta),\ &2^{-4n-4}<\underline{u}\leq 2^{-4n-3}\\ \widetilde{\Psi}^{\prime}_{AB}(\vartheta),\ &2^{-4n-2}<\underline{u}\leq 2^{-4n-1},\\ \text{continuously extended},\ &\text{otherwise}\end{cases}\ \text{for}\ n=0,1,2,\cdots.

The magnitude |Ψ¯|=A,B=1,2(Ψ¯AB)2¯Ψsubscriptformulae-sequence𝐴𝐵12superscriptsubscript¯Ψ𝐴𝐵2|\overline{\Psi}|=\sqrt{\sum_{A,B=1,2}(\overline{\Psi}_{AB})^{2}} is a well-defined function because of the coordinate transformation rule. We then define the normalization ΨABsubscriptΨ𝐴𝐵\Psi_{AB} of Ψ¯ABsubscript¯Ψ𝐴𝐵\overline{\Psi}_{AB} by

ΨAB(u¯,ϑ)=Ψ¯AB(u¯,ϑ)/(1u¯0u¯|Ψ¯|2(u¯,ϑ)du¯)12,subscriptΨ𝐴𝐵¯𝑢italic-ϑsubscript¯Ψ𝐴𝐵¯𝑢italic-ϑsuperscript1¯𝑢superscriptsubscript0¯𝑢superscript¯Ψ2superscript¯𝑢italic-ϑdifferential-dsuperscript¯𝑢12\displaystyle\Psi_{AB}(\underline{u},\vartheta)=\overline{\Psi}_{AB}(\underline{u},\vartheta)\Bigg{/}\left(\frac{1}{\underline{u}}\int_{0}^{\underline{u}}|\overline{\Psi}|^{2}(\underline{u}^{\prime},\vartheta)\mathrm{d}\underline{u}^{\prime}\right)^{\frac{1}{2}}, (6.9)

which is well-defined since 0u¯|Ψ¯|2(u¯,ϑ)du¯superscriptsubscript0¯𝑢superscript¯Ψ2superscript¯𝑢italic-ϑdifferential-dsuperscript¯𝑢\int_{0}^{\underline{u}}|\overline{\Psi}|^{2}(\underline{u}^{\prime},\vartheta)\mathrm{d}\underline{u}^{\prime} is nowhere zero for all u¯(0,1]¯𝑢01\underline{u}\in(0,1]. ΨABsubscriptΨ𝐴𝐵\Psi_{AB} is by definition also a pair of symmetric trace-free matrices satisfying the coordinate transformation rule and for all u¯(0,1]¯𝑢01\underline{u}\in(0,1] and ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2},

1u¯0u¯|Ψ|2(u¯,ϑ)du¯=1.1¯𝑢superscriptsubscript0¯𝑢superscriptΨ2superscript¯𝑢italic-ϑdifferential-dsuperscript¯𝑢1\displaystyle\frac{1}{\underline{u}}\int_{0}^{\underline{u}}|\Psi|^{2}(\underline{u}^{\prime},\vartheta)\mathrm{d}\underline{u}^{\prime}=1. (6.10)

However, ΨΨ\Psi is not continuous at u¯=0¯𝑢0\underline{u}=0 and we will make a cut-off. For any δ>0𝛿0\delta>0, let jδ(u¯)subscript𝑗𝛿¯𝑢j_{\delta}(\underline{u}) be a continuous cut-off function such that jδ(0)=0subscript𝑗𝛿00j_{\delta}(0)=0, jδ(u¯)=1subscript𝑗𝛿¯𝑢1j_{\delta}(\underline{u})=1 for δ2u¯1𝛿2¯𝑢1\frac{\delta}{2}\leq\underline{u}\leq 1, and 0jδ(u¯)10subscript𝑗𝛿¯𝑢10\leq j_{\delta}(\underline{u})\leq 1. Then jδΨABsubscript𝑗𝛿subscriptΨ𝐴𝐵j_{\delta}\Psi_{AB} is defined for all u¯[0,1]¯𝑢01\underline{u}\in[0,1] and from (6.10), we have

1δ0δ|jδΨ|2(u¯,ϑ)du¯12.1𝛿superscriptsubscript0𝛿superscriptsubscript𝑗𝛿Ψ2¯𝑢italic-ϑdifferential-d¯𝑢12\displaystyle\frac{1}{\delta}\int_{0}^{\delta}|j_{\delta}\Psi|^{2}(\underline{u},\vartheta)\mathrm{d}\underline{u}\geq\frac{1}{2}. (6.11)

for all δ(0,1]𝛿01\delta\in(0,1] and ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2}.

Now given any t𝑡t\in\mathbb{R} with small absolute value, we will choose a u~t>εsubscript~𝑢𝑡𝜀\widetilde{u}_{t}>-\varepsilon sufficiently close to 00 according to t𝑡t such that u~t0subscript~𝑢𝑡superscript0\widetilde{u}_{t}\to 0^{-} as t0𝑡0t\to 0. The choice of u~tsubscript~𝑢𝑡\widetilde{u}_{t} will be determined in the course of the proof. We then define

𝔪t;AB(u¯,ϑ)=exp(t0u¯jδ~(u~t;γ)(u¯)ΨAB(u¯,ϑ)du¯)subscript𝔪𝑡𝐴𝐵¯𝑢italic-ϑ𝑡superscriptsubscript0¯𝑢subscript𝑗~𝛿subscript~𝑢𝑡𝛾superscript¯𝑢subscriptΨ𝐴𝐵superscript¯𝑢italic-ϑdifferential-dsuperscript¯𝑢\displaystyle\mathfrak{m}_{t;AB}(\underline{u},\vartheta)=\exp\left(t\int_{0}^{\underline{u}}j_{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}(\underline{u}^{\prime})\Psi_{AB}(\underline{u}^{\prime},\vartheta)\mathrm{d}\underline{u}^{\prime}\right) (6.12)

for all t𝑡t\in\mathbb{R}. Because 0u¯jδ~(u~t;γ)ΨAB(u¯,ϑ)du¯superscriptsubscript0¯𝑢subscript𝑗~𝛿subscript~𝑢𝑡𝛾subscriptΨ𝐴𝐵superscript¯𝑢italic-ϑdifferential-dsuperscript¯𝑢\int_{0}^{\underline{u}}j_{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}\Psi_{AB}(\underline{u}^{\prime},\vartheta)\mathrm{d}\underline{u}^{\prime} is also a pair of symmetric trace-free matrices satisfying the coordinate transformation rule, then 𝔪t;ABsubscript𝔪𝑡𝐴𝐵\mathfrak{m}_{t;AB} is then a pair of symmetric matrices with determinate 111 satisfying the coordinate transformation rule and this defines a family of conformal metrics 𝔤/^t\widehat{\mbox{$\mathfrak{g}\mkern-9.0mu/$}}_{t} through (6.8). We then compute

|Ωχ^𝔤/t|𝔤/t2=14(𝔪t1)AC(𝔪t1)BD𝔪t;ABu¯𝔪t;CDu¯\displaystyle|\Omega\widehat{\chi}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}|^{2}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}=\frac{1}{4}(\mathfrak{m}^{-1}_{t})^{AC}(\mathfrak{m}^{-1}_{t})^{BD}\frac{\partial\mathfrak{m}_{t;AB}}{\partial\underline{u}}\frac{\partial\mathfrak{m}_{t;CD}}{\partial\underline{u}}

in both coordinate charts141414See formula (2.70) in [9].. From the definition (6.12) of 𝔪t;ABsubscript𝔪𝑡𝐴𝐵\mathfrak{m}_{t;AB}, we have

(𝔪t1)AB=IAB+Ot(u¯)superscriptsuperscriptsubscript𝔪𝑡1𝐴𝐵subscript𝐼𝐴𝐵subscript𝑂𝑡¯𝑢\displaystyle(\mathfrak{m}_{t}^{-1})^{AB}=I_{AB}+O_{t}(\underline{u})

where IABsubscript𝐼𝐴𝐵I_{AB} is the identity matrix and

𝔪t;ABu¯=tjδ~(u~t;γ)ΨAB+Ot(u¯).subscript𝔪𝑡𝐴𝐵¯𝑢𝑡subscript𝑗~𝛿subscript~𝑢𝑡𝛾subscriptΨ𝐴𝐵subscript𝑂𝑡¯𝑢\displaystyle\frac{\partial\mathfrak{m}_{t;AB}}{\partial\underline{u}}=tj_{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}\Psi_{AB}+O_{t}(\underline{u}).

Then we have

|Ωχ^𝔤/t|𝔤/t2=t24|jδ~(u~t;γ)Ψ|2+Ot(u¯).\displaystyle|\Omega\widehat{\chi}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}|^{2}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}=\frac{t^{2}}{4}|j_{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}\Psi|^{2}+O_{t}(\underline{u}).

Here the constants in Ot(u¯)subscript𝑂𝑡¯𝑢O_{t}(\underline{u}) depend on Ψ~ABsubscript~Ψ𝐴𝐵\widetilde{\Psi}_{AB}, Ψ~ABsubscriptsuperscript~Ψ𝐴𝐵\widetilde{\Psi}^{\prime}_{AB} and t𝑡t, but not on the particular choice of u~tsubscript~𝑢𝑡\widetilde{u}_{t}. Therefore we can choose u~tsubscript~𝑢𝑡\widetilde{u}_{t} sufficiently close to 00 such that |Ot(u¯)|t216subscript𝑂𝑡¯𝑢superscript𝑡216|O_{t}(\underline{u})|\leq\frac{t^{2}}{16} for 0u¯δ~(u~t;γ)0¯𝑢~𝛿subscript~𝑢𝑡𝛾0\leq\underline{u}\leq\widetilde{\delta}(\widetilde{u}_{t};\gamma) and then from (6.11),

1δ~(u~t;γ)0δ~(u~t;γ)|Ωχ^𝔤/t|𝔤/t2(u¯,ϑ)du¯t216.\displaystyle\frac{1}{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}\int_{0}^{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}|\Omega\widehat{\chi}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}|^{2}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}(\underline{u},\vartheta)\mathrm{d}\underline{u}\geq\frac{t^{2}}{16}. (6.13)

Now we begin to construct g/^t\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}. For the given g/^\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}, we have a pair of matrices mABsubscript𝑚𝐴𝐵m_{AB} given by (6.8). We then define

mt;AB=mAC𝔪t;CBsubscript𝑚𝑡𝐴𝐵subscript𝑚𝐴𝐶subscript𝔪𝑡𝐶𝐵\displaystyle m_{t;AB}=m_{AC}\mathfrak{m}_{t;CB} (6.14)

which also satisfies the coordinate transformation rule and hence defines a family of the conformal metrics g/^t\widehat{\mbox{$g\mkern-9.0mu/$}}_{t} for all small t𝑡t. In particular, g/^0g/^\widehat{\mbox{$g\mkern-9.0mu/$}}_{0}\equiv\widehat{\mbox{$g\mkern-9.0mu/$}} and g/^t\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}\in\mathcal{I} and g/^tg/^\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}\to\widehat{\mbox{$g\mkern-9.0mu/$}} as t0𝑡0t\to 0. Let g/t\mbox{$g\mkern-9.0mu/$}_{t} be the corresponding family of full metrics and χ^tsubscript^𝜒𝑡\widehat{\chi}_{t} be the corresponding family of shear tensors, by direct computations, we have,

|Ωχ^t|g/t2(u¯,u0,ϑ)12|Ωχ^𝔤/t|𝔤/t2|Ωχ^|g/2(u¯,u0,ϑ)Ot(u¯).\displaystyle|\Omega\widehat{\chi}_{t}|_{{\mbox{$g\mkern-9.0mu/$}}_{t}}^{2}(\underline{u},u_{0},\vartheta)\geq\frac{1}{2}|\Omega\widehat{\chi}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}|^{2}_{\mbox{$\mathfrak{g}\mkern-9.0mu/$}_{t}}-|\Omega\widehat{\chi}|_{\mbox{$g\mkern-9.0mu/$}}^{2}(\underline{u},u_{0},\vartheta)-O_{t}(\underline{u}).

By choosing u~tsubscript~𝑢𝑡\widetilde{u}_{t} sufficiently close to zero such that |Ωχ^|g/2(u¯,u0,ϑ)t2128|\Omega\widehat{\chi}|^{2}_{\mbox{$g\mkern-9.0mu/$}}(\underline{u},u_{0},\vartheta)\leq\frac{t^{2}}{128} (recalling that χ^(0,u0,ϑ)=0^𝜒0subscript𝑢0italic-ϑ0\widehat{\chi}(0,u_{0},\vartheta)=0) and |Ot(u¯)|t2128subscript𝑂𝑡¯𝑢superscript𝑡2128|O_{t}(\underline{u})|\leq\frac{t^{2}}{128} for all 0u¯δ~(u~t;γ)0¯𝑢~𝛿subscript~𝑢𝑡𝛾0\leq\underline{u}\leq\widetilde{\delta}(\widetilde{u}_{t};\gamma), ϑS2italic-ϑsuperscript𝑆2\vartheta\in S^{2}, from (6.13), we have

1δ~(u~t;γ)0δ~(u~t;γ)|Ωχ^t|g/t2(u¯,u0,ϑ)du¯t264.\displaystyle\frac{1}{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}\int_{0}^{\widetilde{\delta}(\widetilde{u}_{t};\gamma)}|\Omega\widehat{\chi}_{t}|^{2}_{\mbox{$g\mkern-9.0mu/$}_{t}}(\underline{u},u_{0},\vartheta)\mathrm{d}\underline{u}\geq\frac{t^{2}}{64}. (6.15)

Finally, we can choose u~tsubscript~𝑢𝑡\widetilde{u}_{t} sufficiently close to zero such that Ω02γ(u~t)t264×33superscriptsubscriptΩ02𝛾subscript~𝑢𝑡superscript𝑡26433\Omega_{0}^{2-\gamma}(\widetilde{u}_{t})\leq\frac{t^{2}}{64\times 33}, and conclude that g/^t𝒩2γ,εc\widehat{\mbox{$g\mkern-9.0mu/$}}_{t}\in\mathcal{N}^{c}_{2-\gamma,\varepsilon} for all nonzero small t𝑡t.

 

This proves the denseness of 𝒩2γ,εcsubscriptsuperscript𝒩𝑐2𝛾𝜀\mathcal{N}^{c}_{2-\gamma,\varepsilon} and the proof of Theorem 6.3 is completed.

 

Proof of Theorem 6.4.

We denote

BA={g/^||u0|supu¯[0,1]Ωχ^10(u¯,u0)<A}B_{A}=\left\{\widehat{\mbox{$g\mkern-9.0mu/$}}\in\mathcal{I}\Big{|}|u_{0}|\sup_{\underline{u}\in[0,1]}\|\Omega\widehat{\chi}\|_{\mathbb{H}^{10}(\underline{u},u_{0})}<A\right\}

for every A1𝐴1A\geq 1. Fix a sequence Ai+subscript𝐴𝑖A_{i}\to+\infty, then iBAi=subscript𝑖subscript𝐵subscript𝐴𝑖\bigcup_{i}B_{A_{i}}=\mathcal{I}. From Theorem 6.2, for any given i,ε>0,γ(0,2)formulae-sequence𝑖𝜀0𝛾02i,\varepsilon>0,\gamma\in(0,2), there exists some ε0,i<εsubscript𝜀0𝑖𝜀\varepsilon_{0,i}<\varepsilon such that

𝒩2γ,ε0,icBAiεc.subscriptsuperscript𝒩𝑐2𝛾subscript𝜀0𝑖subscript𝐵subscript𝐴𝑖subscriptsuperscript𝑐𝜀\mathcal{N}^{c}_{2-\gamma,\varepsilon_{0,i}}\cap B_{A_{i}}\subset\mathcal{E}^{c}_{\varepsilon}.

Therefore

εci(𝒩2γ,ε0,icBAi).subscript𝑖subscriptsuperscript𝒩𝑐2𝛾subscript𝜀0𝑖subscript𝐵subscript𝐴𝑖subscriptsuperscript𝑐𝜀\mathcal{E}^{c}_{\varepsilon}\supset\bigcup_{i}(\mathcal{N}^{c}_{2-\gamma,\varepsilon_{0,i}}\cap B_{A_{i}}).

The set on the right is obviously open since it is a union of open subsets, and we only need to show that it is dense. Given any bounded open subset U𝑈U of \mathcal{I}, UBAi0𝑈subscript𝐵subscript𝐴subscript𝑖0U\subset B_{A_{i_{0}}} for some i0subscript𝑖0i_{0}. Then since 𝒩2γ,ε0,i0csubscriptsuperscript𝒩𝑐2𝛾subscript𝜀0subscript𝑖0\mathcal{N}^{c}_{2-\gamma,\varepsilon_{0,i_{0}}} is dense by Proposition 6.1,

Ui(𝒩2γ,ε0,icBAi)U(𝒩2γ,ε0,i0cBAi0)=U𝒩2γ,ε0,i0c.superset-of𝑈subscript𝑖subscriptsuperscript𝒩𝑐2𝛾subscript𝜀0𝑖subscript𝐵subscript𝐴𝑖𝑈subscriptsuperscript𝒩𝑐2𝛾subscript𝜀0subscript𝑖0subscript𝐵subscript𝐴subscript𝑖0𝑈subscriptsuperscript𝒩𝑐2𝛾subscript𝜀0subscript𝑖0U\cap\bigcup_{i}(\mathcal{N}^{c}_{2-\gamma,\varepsilon_{0,i}}\cap B_{A_{i}})\supset U\cap(\mathcal{N}^{c}_{2-\gamma,\varepsilon_{0,{i_{0}}}}\cap B_{A_{i_{0}}})=U\cap\mathcal{N}^{c}_{2-\gamma,\varepsilon_{0,{i_{0}}}}\neq\emptyset.

This shows that εcsubscriptsuperscript𝑐𝜀\mathcal{E}^{c}_{\varepsilon} contains an open and dense subset in \mathcal{I} and completes the proof of Theorem 6.4.

 

References

  • [1] X. An, Formation of trapped surfaces from past null infinity, preprint (2012), arXiv:1207.5271.
  • [2] X. An and J. Luk, Trapped surfaces in vacuum arising dynamically from mild incoming radiation, Advances in Theoretical and Mathematical Physics 21 (2017), 1–120.
  • [3] D. Christodoulou, A Mathematical Theory of Gravitational Collapse, Comm. Math. Phys. 109, (1987), 613–647
  • [4] D. Christodoulou, The formation of black holes and singularities in spherically symmetric gravitational collapse, Communications on Pure and Applied Mathematics 44, no. 3 (1991): 339-373.
  • [5] D. Christodoulou, Bounded variation solutions of the spherically symmetric einstein-scalar field equations, Communications on Pure and Applied Mathematics 46, no. 8 (1993): 1131-1220.
  • [6] D. Christodoulou, Examples of Naked Singularity Formation in the Gravitational Collapse of a Scalar Field, Annals of Mathematics, (1994) 140(3), 607–653.
  • [7] D. Christodoulou, On the global initial value problem and the issue of singularities, Class. Quan. Grav. (1999) 16 A23.
  • [8] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. 149, 183-217 (1999).
  • [9] D. Christodoulou, The Formation of Black Holes in General Relativity, Monographs in Mathematics, European Mathematical Soc. 2009.
  • [10] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of Minkowski Space, Princeton Mathematical Series 41, 1993.
  • [11] M. Dafermos, Spherically symmetric spacetimes with a trapped surface, Classical and Quantum Gravity, 22 (2005), no. 11, 2221
  • [12] R. P. Geroch, E. H. Kronheimer, and R. Penrose, Ideal point in space-time, Proc. Roy. Soc. Lond. Ser. A 327 (1972), 545–567.
  • [13] S. Klainerman, J. Luk, and I. Rodnianski, A fully anisotropic mechanism for formation of trapped surfaces in vacuum, Invent. Math. 198 (2014), no.1, 1–26.
  • [14] S. Klainerman and I. Rodnianski, On the Formation of Trapped Surfaces, Acta Math. 208 (2012), no. 2, 211–333.
  • [15] S. Klainerman, I. Rodnianski and J. Szeftel, The bounded L2superscript𝐿2L^{2} curvature conjecture. Invent. Math., 202 (2015), no. 1, 91–216
  • [16] J. Li and J. Liu, A robust proof of the instability of naked singularities of a scalar field in spherical symmetry, preprint, arXiv:1710.02922.
  • [17] J. Li and X. P. Zhu, On the local extension of the future null infinity, arXiv: 1406.0048, to appear in Journal of Differential Geometry.
  • [18] J. Li and X. P. Zhu, Local existence in retarded time under a weak decay on complete null cones, Sci China Math, 2016, 59: 85–106
  • [19] J. Li and P. Yu, Construction of Cauchy data of vacuum Einstein field equations evolving to black holes, Annals of Mathematics, Volume 181 (2015), Issue 2, 699–768.
  • [20] J. Luk, Weak null singularities in general relativity, preprint (2013), arXiv:1311.4970.
  • [21] J. Luk and I. Rodnianski, Local propagation of impulsive gravitational waves. Comm. Pure Appl. Math., 68 (2015), no. 4, 511–624.
  • [22] J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations preprint (2013), arXiv:1301.1072.
  • [23] A. Rendall, Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. Roy. Soc Lond. A 427 (1990). 221-239
  • [24] P. Yu, Dynamical Formation of black holes due to the condensation of matter field, arXiv:1105.5898.